17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.
y'(t) = yeᵗ, y(0) = −1

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First, identify if the differential equation is separable. The given equation is \(y'(t) = y e^{t}\). Since the right side can be written as a product of a function of \(y\) and a function of \(t\), it is separable.

Rewrite the differential equation using Leibniz notation: \(\frac{dy}{dt} = y e^{t}\). Then separate variables by dividing both sides by \(y\) and multiplying both sides by \(dt\): \(\frac{1}{y} dy = e^{t} dt\).

Integrate both sides: \(\int \frac{1}{y} dy = \int e^{t} dt\). This will give you the antiderivatives on each side.

After integrating, you will have \(\ln|y| = e^{t} + C\), where \(C\) is the constant of integration. Solve for \(y\) by exponentiating both sides: \(y = \pm e^{e^{t} + C} = A e^{e^{t}}\), where \(A = \pm e^{C}\).

Use the initial condition \(y(0) = -1\) to find the constant \(A\). Substitute \(t=0\) and \(y=-1\) into the solution and solve for \(A\). Then write the particular solution.

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Separable Differential Equations

A differential equation is separable if it can be written as a product of a function of y and a function of t, allowing the variables to be separated on opposite sides of the equation. This form enables integration with respect to each variable independently to find the solution.

Initial Value Problems (IVP)

An initial value problem involves solving a differential equation with a given initial condition, such as y(t₀) = y₀. The initial condition helps determine the specific solution curve among the family of solutions by fixing the constant of integration.

Integration Techniques for Solving ODEs

Once variables are separated, integration is used to solve each side with respect to its variable. This often involves integrating exponential functions or polynomials, and applying the initial condition to solve for the integration constant to find the explicit solution.

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.y'(t) = yeᵗ, y(0) = −1