How do you use exponent rules to separate variables in the differential equation y' = $e^{2x-y}$ ?

First, apply the quotient rule for exponents with the same base to rewrite

e^{2x-y}

e^{2x}/e^y

. Replace y' with

dy/dx

to get

dy/dx = e^{2x}/e^y

. Then, multiply both sides by

e^y dx

to move all y terms to the left and all x terms to the right, resulting in

e^y dy = e^{2x} dx

Conditions

The equation is a first-order differential equation.
The exponent rule $a^m/a^n = a^{m-n}$ is applicable.

Reasoning, step by step

Start with the equation: y' = $e^{2x-y}$ .
Rewrite y' as $dy/dx$ : $dy/dx = e^{2x-y}$ .
Apply the exponent rule to split the right side: $dy/dx = e^{2x}/e^y$ .
Multiply both sides by $e^y dx$ to separate the variables.
Obtain the final separable form: $e^y dy = e^{2x} dx$ .

Example

The video demonstrates this exact sequence, using the exponent rule to break apart the composite exponential term before separating the differentials.

Common misconceptions

Incorrectly applying the exponent rule, such as writing $e^{2x-y}$ as $e^{2x} - e^y$ .
Forgetting to multiply by dx when moving the differential term.

Watch the explanation

BilibiliDifferential equations: how to quickly distinguish Separable equations, Linear equations, and differential equations with variables swapped about X

4:30 – 6:00Watch this moment ↗

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Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.

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