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Substitute the trial solution $x=Ate^{-t}$ and its derivatives into the ODE, factor out the common term $e^{-t}$, and collect like terms by powers of $t$. Equating the remaining constant term to the right-hand side yields $A=2$.

Conditions: The trial solution is $x=Ate^{-t}$.; The ODE is $\ddot{x}+3\dot{x}+2x=2e^{-t}$.; $e^{-t}$ is never zero, allowing it to be canceled from both sides.

The time average of an observable $J$ along a trajectory $u(t)$ is defined as the limit of the average of $J$ over a finite time window of length $T$ as $T$ approaches infinity. It represents the long-run average behavior of the observable.

Conditions: The system is described by an ODE with trajectory $u(t)$.; $J$ is a function of the state $u$.; The averaging interval starts at $t=0$ and has length $T$.; The limit $T\to\infty$ is taken.

The decision rule is based on comparing the class average to a fixed threshold. An outlier artificially inflates the average above the threshold, preventing the class from receiving help.

Conditions: The decision rule is: hold extra classes only if the average score is below 35.; The dataset contains an outlier.; The threshold is fixed at 35.

To calculate the future value (FV) of a lump sum, you use the compound interest formula $FV = PV(1+r)^n$. You substitute the present value (PV), the interest rate per period (r), and the number of periods (n) into the formula.

Conditions: The interest rate r is assumed to remain constant over the period.; The compounding frequency matches the time unit of n (e.g., annual rate with years).; The formula assumes a single lump-sum payment rather than an annuity.

A vector field over $\mathbb{R}^3$ assigns to each point $p$ a vector $v$ belonging to the tangent space $T_p\mathbb{R}^3$ at that point.

Conditions: The ambient space in this example is $\mathbb{R}^3$.

The percent symbol '%' is a shorthand notation used to represent a fraction with a denominator of 100. When placed after a number, it indicates that the number is the numerator of a fraction over 100.

Conditions: Used immediately after a numerical value.; Represents a fraction with a denominator of 100.

To find nonnegative integer solutions for an equation like $x_1+\cdots+x_n=r$ where one or more variables have a lower bound greater than zero, substitute a new variable to shift that bound to zero. For instance, if $x_3 \ge 2$, define $x_3' = x_3 - 2$ and subtract the lower bound from the total sum $r$. This transforms the problem into finding nonnegative solutions for the adjusted equation, which can then be solved using the standard stars and bars method.

Conditions: One or more variables have a lower bound greater than zero.; The other variables are nonnegative.; The equation is of the form $x_1+\cdots+x_n=r$.

The directional derivative in the direction of a vector w is equal to the dot product of w and the gradient vector ∇f. The gradient vector contains the partial derivatives of the function as its components.

Conditions: The function $f(x,y)$ is differentiable.; The direction vector is $w=[a,b]$.; The gradient ∇f = [∂f/∂x, ∂f/∂y] exists.

The Year 2 forecast is 310 because both the actual demand for Year 1 and the initialized forecast for Year 1 are exactly 310. When a weighted average is calculated using two identical values, the result is that same value, regardless of the weights used.

Conditions: The smoothing constant $\alpha$ is fixed at 0.4.; The actual demand for Year 1 ($A_1$) is 310.; The initialized forecast for Year 1 ($F_1$) is 310.

An affine scheme is a locally ringed space that is isomorphic to the spectrum Spec(R) of a commutative ring R. General schemes are constructed by gluing together open sets of this simple kind, called affine schemes.

Conditions: The object is an affine scheme.; It is a locally ringed space.; It is isomorphic to the spectrum of a commutative ring.

To verify an ordered pair solves a linear equation with fractions, substitute the $x$ and $y$ values into the left-hand side of the equation. Evaluate the products by multiplying each fraction by its corresponding integer.

Conditions: The candidate solution values must already be known.; The equation being checked contains those variables with fractional coefficients.

The iteration threshold $\gamma_i$ is defined as the empirical $(1-p)$-quantile of the current batch of samples drawn from the proposal distribution. The loop stops when this threshold $\gamma_i$ is greater than or equal to the target rare-event threshold $\gamma$.

Conditions: Samples are drawn from the current proposal distribution.; The elite fraction parameter $p$ is set to 0.05.; The target threshold is $\gamma=5$.