Modeling population with simple differential equation | Khan Academy

1. Audio
   Observation

   The speaker says, "Let's say that P is equal to our population."

2. Formula
   Observation

   The board shows the handwritten definition "P = population".

P

Population variable in the model.

Population count (implicitly nonnegative).

1. Audio
   Observation

   The speaker says, "let's say that t is equal to the time that has passed in days."

2. Formula
   Observation

   The board shows the handwritten definition "t = time (days)".

t

Time elapsed, measured in days.

Time in days.

1. Audio
   Observation

   The speaker says, "the rate of change of our population with respect to time" and later refers to "the derivative of P with respect to T".

2. Formula
   Observation

   The board shows the handwritten fraction-like notation "dP/dt" on the left side of the equation.

dP/dt

Instantaneous rate of change of population with respect to time.

Derivative of $P(t)$ with respect to t.

1. Audio
   Observation

   The speaker says the rate of change is "proportional to the actual population," which implies a proportionality constant, but does not explicitly name it aloud.

2. Formula
   Observation

   The board shows an orange handwritten constant symbol before P in the equation, visually consistent with k.

1. The constant is written on the board but not verbally named in the clip; identification as k is based on the visible symbol and standard proportionality notation.

k

Proportionality constant relating dP/dt to P.

Constant multiplier in the differential equation.

1. Formula
   Observation

   The board writes "P = population" and uses P in the differential equation and later integrals.

P

Population variable.

Population quantity; treated as a function of time t.

1. Formula
   Observation

   The board writes "t = time (days)" and uses t in dP/dt and kt.

t

Time measured in days.

Independent variable for the differential equation.

1. Formula
   Observation

   The equation dP/dt = kP is written on the board, and k remains in the separated and integrated forms.

1. The video does not explicitly state the sign or units of k.

k

Constant proportionality factor in the population model.

Constant parameter multiplying P in dP/dt = kP.

1. Formula
   Observation

   The board shows dP/dt in Leibniz notation.

2. Audio
   Observation

   The speaker says to treat this derivative in Leibniz notation like fractions for separable differential equations.

dP/dt

Instantaneous rate of change of population with respect to time.

Derivative notation for P as a function of t.

1. Formula
   Observation

   After integration, the board writes + $C_1$ on the right-hand side.

2. Audio
   Observation

   The speaker says, "I'll just call that C one."

$C_1$

Integration constant introduced after integrating both sides.

Arbitrary constant.

1. Formula
   Observation

   The integrated left side is written as ln|P|.

2. Audio
   Observation

   The speaker says the integral gives the natural log of the absolute value of our population.

ln|P|

Natural logarithm of the absolute value of P.

Logarithmic expression arising from $\int (1/P)$dP.

1. Formula
   Observation

   The board rewrites |P| = e^{kt+$C_1$} and then e^{kt}$e^{C_1}$.

2. Audio
   Observation

   The speaker says the absolute value of P is equal to e to all of this business.

e

Base of the natural exponential used to undo the natural logarithm.

Positive constant base for exponentiation.

1. Formula
   Observation

   P = population

P

population

positive real numbers (assumed in the video)

1. Audio
   Observation

   The speaker introduces the goal as "start exploring how we can model things with differential equations" and specifically "modeling population."

2. Formula
   Observation

   The board defines "P = population" and "t = time (days)".

The clip begins by framing differential equations as a modeling tool. The presenter defines the dependent variable P as the population and the independent variable t as elapsed time measured in days, choosing days because the example concerns insects that reproduce quickly.

1. The model is for population change over time.

2. Time is measured in days in this setup.

1. Audio
   Observation

   The speaker says, "the rate of change of our population with respect to time ... is going to be proportional to the actual population."

2. Formula
   Observation

   The board shows the handwritten equation "dP/dt = kP".

The central model in this clip is the differential equation dP/dt = kP. It states that the instantaneous rate at which the population changes is proportional to the current population size.

1. P denotes population.

2. t denotes time in days.

3. k is the proportionality constant shown on the board.

1. Setting up a population model with variables P and t

1. Audio
   Observation

   The speaker says, "you might immediately recognize that this is a separable differential equation."

2. Audio
   Observation

   He explains that in separable equations, one wants one variable and its differentials on one side and the other variable and its differentials on the other side, then integrate both sides.

After writing the model, the presenter identifies it as separable. The stated method is to rearrange so that all terms involving P and dP are on one side and all terms involving t and dt are on the other, then integrate both sides.

1. The equation must be rearrangeable into separated variables before integration.

1. Differential equation for proportional population growth

1. Formula
   Observation

   The title reads "Modeling Population" and the board defines P = population, t = time (days), with dP/dt = kP.

2. Audio
   Observation

   The speaker discusses solving this differential equation by separation of variables.

1. The clip does not explain why dP/dt = kP is the chosen population law beyond treating it as the starting model.

The clip models population growth with the differential equation dP/dt = kP, where P is population and t is time in days. The equation is then solved by separating the P-dependent terms from the t-dependent terms and integrating both sides.

1. P denotes population.

2. t denotes time in days.

3. k is a constant parameter.

1. Audio
   Observation

   The speaker says to put all the P's and dP's on one side and the things involving t on the other side.

2. Formula
   Observation

   The board transforms dP/dt = kP into ($1/P$)(dP/dt)=kP($1/P$), then multiplies by dt, yielding ($1/P$)dP = k dt.

For this equation, the method is to divide by P and multiply by dt so that all P-related quantities sit on the left and all t-related quantities sit on the right. The speaker explicitly frames this as valid for separable differential equations when using Leibniz notation heuristically.

1. The differential equation must be separable.

2. The manipulation treats dP and dt algebraically as a working device before integration.

1. Population model as a separable differential equation

1. Formula
   Observation

   The board writes $\int (1/P)$dP = $\int k dt$ and then ln|P| = kt + $C_1$.

2. Audio
   Observation

   The speaker says, "Now we can integrate both sides," and explains that the left side becomes the natural log of the absolute value of P.

Once the equation is separated, each side is integrated. The left integral $\int (1/P)$dP produces ln|P|, while the right integral $\int k dt$ produces kt plus an arbitrary constant. The speaker notes that only one constant needs to be kept because constants on both sides can be combined.

1. $P \ne 0$ for division by P.

2. k is constant with respect to t.

1. Separating variables in a first-order ODE

1. Formula
   Observation

   The board rewrites ln|P| = kt + $C_1$ as |P| = e^{kt+$C_1$}, then as |P| = e^{kt}$e^{C_1}$.

2. Audio
   Observation

   The speaker asks how to solve for P and says the natural log equation means the absolute value of P equals e to the whole expression.

1. The clip ends before the final simplification to P = Ce^{kt} is written out completely.

To isolate P from ln|P| = kt + $C_1$, exponentiate both sides using the inverse relationship between ln and e. This gives |P| = e^{kt+$C_1$}. Then exponent rules split the expression into |P| = e^{kt}$e^{C_1}$, setting up the usual exponential solution form.

1. Exponentiation is applied to both sides of the logarithmic equation.

2. $e^{C_1}$ is positive because it is an exponential of a real constant.

1. Integrating both sides after separation

1. Formula
   Observation

   $dt \frac{1}{P} \frac{dP}{dt}$ = kP $\cdot \frac{1}{P} \cdot dt$

To solve the differential equation, both sides are multiplied by dt and divided by P to separate the variables P and t.

1. $P \neq 0$

1. Formula
   Observation

   $\int \frac{1}{P}$ dP = $\int k dt$

Integrating both sides of the separated equation yields the natural logarithm of the absolute value of P on the left and kt plus a constant on the right.

1. Separation of Variables

1. Formula
   Observation

   |P| = e^{kt+$C_1$} = e^{kt}$e^{C_1}$

Applying the exponential function to both sides removes the natural logarithm, and the properties of exponents are used to split the term.

1. Integration of Separated Equation

1. Audio
   Observation

   And if we assume our population at any given time is positive, then we could get rid of this absolute value sign.

Since population is assumed to be positive, the absolute value signs can be removed.

1. $P > 0$

1. Exponentiating Both Sides

1. Audio
   Observation

   Now this is just e to some constant so we could just call this... let's just call that the constant C.

The term $e^{C_1}$ is a constant, which is renamed as C to simplify the general solution.

1. Removing Absolute Value

1. Audio
   Observation

   The speaker asks why the model is reasonable and answers that a larger population gives a larger rate at any given time, comparing 1000 insects with 10 insects.

If the population is larger, then the reproduction rate at a given time is larger, so modeling dP/dt as proportional to P is reasonable.

1. The population consists of reproducing individuals such as insects.

2. Rate of change is considered at a given time.

For the population sizes compared in the example, larger population corresponds to larger total reproduction rate.

1. Audio
   Observation

   The speaker says, "the derivative of P with respect to T, this isn't quite a fraction. This is the limit as our change in P over change in time."

The notation dP/dt should be understood as an instantaneous rate of change defined by a limit, not literally as an ordinary fraction.

1. Standard calculus interpretation of derivatives is assumed.

Universal within the clip's explanation of derivative notation.

1. Audio
   Observation

   The speaker says that for separable differential equations you can treat the derivative in Leibniz notation like fractions and treat differentials like quantities because they will eventually be integrated.

1. The statement is presented as a practical rule for solving separable equations rather than as a rigorous definition of differentials.

In separable differential equations, dP/dt in Leibniz notation can be manipulated heuristically like a fraction, and the differentials can be treated like quantities for the purpose of separation and later integration.

1. The differential equation is separable.

2. The notation being used is Leibniz notation.

For separable differential equations in this context.

1. Audio
   Observation

   The speaker says the differential really isn't a quantity and that one has to view this as the limit of change in P over change in time as changes in time get smaller and smaller.

Although the algebraic manipulation treats dP and dt like quantities, the derivative should really be understood as the limit of change in P over change in time as the time increment shrinks.

1. The object under discussion is dP/dt.

Conceptual statement about derivative notation.

1. Audio
   Observation

   The speaker says one could add a constant on the left too, but then subtract constants from both sides and end up with a single constant on the right.

2. Formula
   Observation

   Only + $C_1$ appears on the right side after integration.

When integrating both sides of a separated differential equation, it is sufficient to write one arbitrary constant on one side because constants from both sides can be combined into a single constant.

1. Both sides are being integrated indefinitely.

For indefinite integration of both sides in this solution procedure.

1. Audio
   Observation

   The speaker moves from asking what a reasonable model would be to stating that the rate of change is proportional to the population.

2. Formula
   Observation

   The board writes dP/dt = kP after the verbal setup.

1. Expression

   $$\text{Choose } P=\text{population},\ t=\text{time in days}.$$
   Explanation

   The model first assigns meanings to the variables.

   Justification

   Explicitly stated by the speaker and written on the board.

   Shown in the video
2. Expression

   $$\frac{dP}{dt} \propto P$$
   Explanation

   The speaker proposes that the population's instantaneous rate of change is proportional to the current population.

   Justification

   Direct verbal modeling assumption.

   Shown in the video
3. Expression

   $$\frac{dP}{dt}=kP$$
   Explanation

   The proportionality statement is written as an equation using a constant multiplier.

   Justification

   Board notation converts the verbal proportionality into a differential equation.

   Shown in the video

The simple population model in this clip is the differential equation dP/dt = kP.

1. Audio
   Observation

   The speaker says they can try to solve the differential equation, find a general solution, then use initial conditions or known population states to solve for constants and find a particular solution.

2. Audio
   Observation

   He then identifies the equation as separable and describes separating variables before integrating both sides.

1. The clip stops before the actual separation step and before any integration is carried out.

1. Expression

   $$\frac{dP}{dt}=kP$$
   Explanation

   Start from the modeled differential equation.

   Justification

   This is the equation already written on the board.

   Shown in the video
2. Expression

   $$\text{Separate } P\text{-terms and } t\text{-terms.}$$
   Explanation

   The speaker says a separable equation is handled by putting one variable and its differentials on one side and the other variable and its differentials on the other side.

   Justification

   Explicit verbal description of the separable-equation method.

   Shown in the video
3. Expression

   $$\text{Integrate both sides.}$$
   Explanation

   After separation, the next stated step is integration on both sides.

   Justification

   Explicitly said by the speaker.

   Shown in the video
4. Expression

   $$\text{Use initial conditions to determine constants.}$$
   Explanation

   The speaker says that after finding a general solution, one can impose initial conditions or known population states to obtain a particular solution.

   Justification

   Explicit verbal plan.

   Shown in the video

The clip outlines the solution strategy—recognize separability, separate variables, integrate, then apply initial conditions—but does not execute those steps within the provided duration.

1. Formula
   Observation

   The board successively shows dP/dt = kP, then ($1/P$)(dP/dt)=kP($1/P$), then multiplication by dt, then ($1/P$)dP = k dt, then $\int (1/P)$dP = $\int k dt$, then ln|P| = kt + $C_1$.

2. Audio
   Observation

   The speaker narrates dividing by P, multiplying by dt, and integrating both sides.

1. Expression

   $$\frac{dP}{dt}=kP$$
   Explanation

   Start from the population differential equation written on the board.

   Justification

   Given model.

   Shown in the video
2. Expression

   $$\frac{1}{P}\frac{dP}{dt}=kP\cdot\frac{1}{P}$$
   Explanation

   Divide both sides by P so the P-dependence can be isolated on the left.

   Justification

   Algebraic manipulation of both sides of an equation.

   Shown in the video
3. Expression

   $$\frac{1}{P}\,dP=k\,dt$$
   Explanation

   Multiply by dt and cancel the P factors on the right, leaving all P terms on the left and t terms on the right.

   Justification

   Heuristic use of Leibniz notation for separable equations, as stated by the speaker.

   Shown in the video
4. Expression

   $$\int \frac{1}{P}\,dP=\int k\,dt$$
   Explanation

   Integrate both sides after separation.

   Justification

   Standard step for solving a separable differential equation.

   Shown in the video
5. Expression

   $$\ln|P|=kt+C_1$$
   Explanation

   Evaluate the integrals: the left side gives the natural log of absolute P, and the right side gives kt plus a constant.

   Justification

   Antiderivative of $1/P$ is ln|P|; antiderivative of a constant k with respect to t is kt.

   Shown in the video

The separated and integrated form of the model is ln|P| = kt + $C_1$.

1. Formula
   Observation

   The board rewrites ln|P| = kt + $C_1$ as |P| = e^{kt+$C_1$}, then as |P| = e^{kt}$e^{C_1}$.

2. Audio
   Observation

   The speaker says this is the same thing using exponent properties.

1. The final absorption of $e^{C_1}$ into a new constant is not completed within the clip.

1. Expression

   $$\ln|P|=kt+C_1$$
   Explanation

   Begin with the logarithmic equation obtained after integration.

   Justification

   Previous derivation result.

   Shown in the video
2. Expression

   $$|P|=e^{kt+C_1}$$
   Explanation

   Exponentiate both sides to remove the natural logarithm.

   Justification

   The exponential function is the inverse of the natural logarithm.

   Shown in the video
3. Expression

   $$|P|=e^{kt}e^{C_1}$$
   Explanation

   Split the exponential of a sum into a product of exponentials.

   Justification

   Exponent rule $e^{a+b}=e^a e^b$.

   Shown in the video

The solution is rewritten in exponential form as |P| = e^{kt}$e^{C_1}$, preparing for the standard exponential population formula.

1. Formula
   Observation

   Step-by-step derivation from $dt \frac{1}{P} \frac{dP}{dt}$ = kP $\cdot \frac{1}{P} \cdot dt$ to $P = C e$^{kt}

1. Expression

   $$\frac{1}{P} dP = k dt$$
   Explanation

   Separate the variables P and t.

   Justification

   Algebraic manipulation of the differential equation.

   Shown in the video
2. Expression

   $$\ln|P| = kt + C_1$$
   Explanation

   Integrate both sides.

   Justification

   Fundamental theorem of calculus and integration rules.

   Shown in the video
3. Expression

   $$|P| = e^{kt+C_1} = e^{kt}e^{C_1}$$
   Explanation

   Exponentiate both sides and use exponent rules.

   Justification

   Properties of logarithms and exponents.

   Shown in the video
4. Expression

   $$P = e^{kt}e^{C_1}$$
   Explanation

   Remove absolute value based on the assumption that population is positive.

   Justification

   Assumption $P > 0$.

   Shown in the video
5. Expression

   $$P = C e^{kt}$$
   Explanation

   Substitute C for $e^{C_1}$.

   Justification

   Definition of a new constant.

   Shown in the video

The general solution to the differential equation is $P = C e$^{kt}.

1. Audio
   Observation

   The speaker says the time unit could have been years or months, but they are considering insects that reproduce quickly, so days are convenient.

2. Audio
   Observation

   He compares 1000 insects with 10 insects to justify proportionality.

Use a simple differential-equation model to describe how an insect population changes over time.

1. P is the population.

2. t is time in days.

3. The population being modeled is insects that reproduce quickly.

Formulate a reasonable differential equation for population change.

1. Expression

   $$\text{Choose days as the time unit.}$$
   Explanation

   Because insects reproduce quickly, days are a convenient scale for observing change.

   Justification

   Stated directly by the speaker.

   Shown in the video
2. Expression

   $$\frac{dP}{dt}=kP$$
   Explanation

   Model the rate of population change as proportional to the current population.

   Justification

   Justified verbally by comparing larger and smaller populations and written on the board.

   Shown in the video

The motivating example leads to the model dP/dt = kP with t measured in days.

The speaker verifies reasonableness intuitively by arguing that 1000 insects produce more new individuals per unit time than 10 insects.

1. Formula
   Observation

   The entire clip works through the example equation dP/dt = kP under the heading Modeling Population.

2. Audio
   Observation

   The speaker solves the equation step by step by separation, integration, and exponentiation.

1. No numerical initial condition is provided, so no specific population curve is determined.

Solve the separable differential equation dP/dt = kP, where P is population and t is time in days.

1. P = population

2. t = time (days)

3. dP/dt = kP

Find an explicit relation for P in terms of t.

1. Expression

   $$\frac{1}{P}\frac{dP}{dt}=kP\cdot\frac{1}{P}$$
   Explanation

   Divide by P to begin separating variables.

   Justification

   Algebraic manipulation of the given equation.

   Shown in the video
2. Expression

   $$\frac{1}{P}\,dP=k\,dt$$
   Explanation

   Multiply by dt so P terms and t terms are on opposite sides.

   Justification

   Separable-equation method using Leibniz notation heuristically.

   Shown in the video
3. Expression

   $$\int \frac{1}{P}\,dP=\int k\,dt$$
   Explanation

   Integrate both sides.

   Justification

   Standard solution step after separation.

   Shown in the video
4. Expression

   $$\ln|P|=kt+C_1$$
   Explanation

   Compute the antiderivatives.

   Justification

   $\int (1/P)$dP = ln|P| and $\int k dt$ = kt + constant.

   Shown in the video
5. Expression

   $$|P|=e^{kt+C_1}$$
   Explanation

   Exponentiate both sides to solve for |P|.

   Justification

   Inverse relationship between ln and e.

   Shown in the video
6. Expression

   $$|P|=e^{kt}e^{C_1}$$
   Explanation

   Use exponent rules to separate the constant factor.

   Justification

   $e^{a+b}=e^a e^b$.

   Shown in the video

The clip reaches |P| = e^{kt}$e^{C_1}$; it does not finish rewriting this as P = Ce^{kt}.

Substitution back into the original differential equation is not shown in the clip.

1. Animation
   Observation

   A digital blackboard is used throughout the clip.

2. Formula
   Observation

   The title "Modeling Population" appears first, followed by the definitions "P = population" and "t = time (days)", then the equation "dP/dt = kP".

1. Title text "Modeling Population"

2. Variable definition "P = population"

3. Variable definition "t = time (days)"

4. Equation "dP/dt = kP"

5. Cursor/pen movements

1. The board starts blank.

2. The title is written first.

3. The variable definitions are added beneath the title.

4. The differential equation is written last, with the constant shown in orange.

1. The setting remains a black digital board with handwritten text.

2. No graph or numerical table is introduced in this clip.

The visual progression mirrors the mathematical build-up: name the topic, define variables, then state the governing differential equation.

1. Diagram
   Observation

   A black digital whiteboard shows the pink title "Modeling Population", the definitions P = population and t = time (days), and the equation dP/dt = kP.

1. Title "Modeling Population"

2. Definitions P = population and t = time (days)

3. Equation dP/dt = kP

1. The board presents the model before any algebraic manipulation begins.

1. The variables P and t keep the same meanings throughout the clip.

2. The equation starts in the form dP/dt = kP.

The visual layout establishes the modeling context and the differential equation to be solved.

1. Animation
   Observation

   New lines are written beneath the original equation as the speaker divides by P, cancels P on the right, and multiplies by dt.

2. Formula
   Observation

   The intermediate expressions ($1/P$)(dP/dt)=kP($1/P$) and then ($1/P$)dP = k dt appear on the board.

1. Original equation dP/dt = kP

2. Fraction $1/P$

3. Differential dt

4. Separated equation ($1/P$)dP = k dt

1. P is moved to the left side through division.

2. dt is introduced by multiplication.

3. The right side simplifies after cancellation of P.

1. k remains a constant multiplier on the t-side.

2. The equation remains an equality throughout the manipulation.

The writing sequence visually demonstrates the separation step that makes integration possible.

1. Animation
   Observation

   Integral signs are added to both sides, then the board rewrites the result as ln|P| = kt + $C_1$.

2. Formula
   Observation

   The integrated equation appears below the separated form.

1. $\int (1/P)$dP

2. $\int k dt$

3. ln|P|

4. kt + $C_1$

1. The separated differential form is converted into an integrated equation.

2. A constant $C_1$ is appended on the right side.

1. The left side depends only on P; the right side depends only on t and a constant.

This visual transition shows how separation leads directly to antiderivatives and a logarithmic relation.

1. Animation
   Observation

   The board writes |P| = e^{kt+$C_1$}, then rewrites it as |P| = e^{kt}$e^{C_1}$.

2. Formula
   Observation

   Exponential notation replaces the logarithmic equation.

1. The clip stops before the final constant renaming step is fully written.

1. |P|

2. e^{kt+$C_1$}

3. e^{kt}$e^{C_1}$

1. The logarithmic equation is transformed into an exponential equation.

2. The exponent sum is split into a product using exponent rules.

1. The absolute value bars around P remain visible in the shown steps.

The visual sequence demonstrates how to isolate P from a logarithmic expression and prepare the familiar exponential growth form.

1. Animation
   Observation

   The instructor writes the steps of the derivation on the blackboard, culminating in the boxed final equation $P = C e$^{kt}.

1. blackboard

2. chalk

3. equations

1. Equations are written sequentially from top to bottom.

2. The final equation is enclosed in a box.

1. The initial differential equation remains visible at the top.

Visual representation of the algebraic and calculus steps used to solve the differential equation.

1. Audio
   Observation

   The speaker says, "sometimes you think of differential equations as these daunting complex things, but notice we've just been able to express a reasonably not so complicated idea."

Differential equations may seem inherently daunting or overly complicated.

The clip shows that a simple real-world idea—population growth rate proportional to population size—can be expressed compactly as dP/dt = kP.

1. Audio
   Observation

   The speaker says, "this isn't quite a fraction. This is the limit as our change in P over change in time."

The derivative notation dP/dt can be mistaken for a literal fraction.

The presenter emphasizes that dP/dt represents an instantaneous rate of change defined through a limit, even though separable-equation manipulations use fraction-like notation.

1. Audio
   Observation

   The speaker explicitly warns that treating the differential like a quantity is only for the sake of the method; it really isn't a quantity and should be viewed as a limit.

One may think dP and dt are ordinary algebraic quantities in themselves.

The clip states that this treatment is a useful procedural device for separable equations, while the derivative itself is fundamentally a limit of change in P over change in time.

1. Audio
   Observation

   The speaker says one could add a constant on the left too, but then subtract constants from both sides and end up with a single constant on the right.

After integrating both sides, one might believe both sides require distinct constants that must remain separate.

The clip explains that constants from both sides can be combined, so writing one constant such as $C_1$ on one side is sufficient.

1. Formula
   Observation

   The board first defines P and t, then writes dP/dt = kP.

2. Audio
   Observation

   The speaker introduces the variables before proposing the proportional model.

Defining P as population and t as time is required before interpreting dP/dt and writing the population-growth differential equation.

1. Audio
   Observation

   After presenting dP/dt = kP, the speaker says it is a separable differential equation and explains the separation-and-integration approach.

The separable-equation method is applied to the specific model dP/dt = kP.

1. Audio
   Observation

   The speaker warns that dP/dt is not quite a fraction while discussing separable equations.

The clip contrasts the formal meaning of the derivative with the practical fraction-like manipulation used when separating variables.

1. Formula
   Observation

   The clip begins with dP/dt = kP and immediately rewrites it into separated form.

2. Audio
   Observation

   The speaker says they want to put all the P's and dP's on one side and the t-related terms on the other.

The population differential equation is solved by applying the separation-of-variables method.

1. Formula
   Observation

   After obtaining ($1/P$)dP = k dt, the board adds integral signs to both sides.

2. Audio
   Observation

   The speaker says, "Now we can integrate both sides."

The integration step depends on first having separated the variables into P-only and t-only sides.

1. Formula
   Observation

   The logarithmic result ln|P| = kt + $C_1$ is rewritten as |P| = e^{kt+$C_1$}.

2. Audio
   Observation

   The speaker asks how to solve for P and answers by exponentiating.

Solving explicitly for P requires exponentiating the logarithmic equation produced by integration.

1. Audio
   Observation

   The speaker contrasts treating dP/dt like a fraction with the deeper view of the derivative as a limit.

The clip contrasts the practical fractional manipulation of Leibniz notation with the conceptual limit definition of the derivative.

1. Audio
   Observation

   this is just an exponential function. And it's very likely that in algebra or in pre-calculus class you have modeled things with exponential functions

The final solution $P = C e$^{kt} is identified as an exponential function, connecting the calculus derivation to prior algebra knowledge.

1. Formula
   Observation

   Board text defines P = population.

1. Setting up a population model with variables P and t

1. Audio
   Observation

   The speaker explains that days are chosen because the example involves insects that reproduce quickly.

1. Setting up a population model with variables P and t
2. Insect population as the motivating example

1. Audio
   Observation

   The speaker justifies proportionality by comparing larger and smaller populations.

2. Formula
   Observation

   The equation dP/dt = kP is written on the board.

1. Differential equation for proportional population growth
2. Why proportionality to population is a reasonable modeling assumption

1. Audio
   Observation

   The speaker says the equation is separable and describes separating variables and integrating both sides.

1. The actual algebraic separation and integration are not shown in this clip.

1. Recognizing dP/dt = kP as a separable differential equation
2. Planned method for solving the model

1. Audio
   Observation

   The speaker explicitly says dP/dt is not quite a fraction and is a limit.

1. Interpretation of dP/dt
2. Treating dP/dt as an ordinary fraction

1. Audio
   Observation

   The speaker explains why Leibniz notation can be treated like fractions in separable differential equations.

1. Population model as a separable differential equation
2. Separating variables in a first-order ODE

1. Formula
   Observation

   The board divides by P and cancels P on the right side.

1. Separating variables in a first-order ODE

1. Audio
   Observation

   The speaker explains that constants from both sides can be combined into one constant on the right.

1. Integrating both sides after separation
2. Combining integration constants into one constant

1. Formula
   Observation

   The board converts ln|P| = kt + $C_1$ into |P| = e^{kt+$C_1$} and then |P| = e^{kt}$e^{C_1}$.

1. Solving for P by exponentiating

1. Audio
   Observation

   The speaker says the derivative should really be viewed as the limit of change in P over change in time.

1. Derivative as a limit rather than a literal quotient

1. Formula
   Observation

   Derivation of $P = C e$^{kt}

1. Separation of Variables
2. Integration of Separated Equation
3. Exponentiating Both Sides
4. Removing Absolute Value
5. Defining a New Constant