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

Verified step by step guidance

1. First, rewrite the differential equation in the form \( \frac{dy}{dt} = e^{ty} \) to clearly identify the variables involved.
2. Check if the equation is separable by trying to express it as a product of a function of \( t \) and a function of \( y \), i.e., \( \frac{dy}{dt} = g(t)h(y) \). In this case, observe that \( e^{ty} \) cannot be separated into a product of a function of \( t \) alone and a function of \( y \) alone.
3. Since the equation is not separable, the standard method of separation of variables does not apply here. Consider alternative methods such as substitution or recognizing the equation type.
4. If you attempt substitution, for example, let \( u = ty \), then express \( y \) and \( y' \) in terms of \( u \) and \( t \) to transform the equation into a potentially separable or solvable form.
5. After substitution, solve the resulting differential equation for \( u(t) \), then back-substitute to find \( y(t) \). Finally, apply the initial condition \( y(0) = 1 \) to determine the constant of integration.

Key Concepts

Separable Differential Equations

A differential equation is separable if it can be written as a product of a function of t and a function of y, 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 specifies the value of the unknown function at a particular point, providing a unique solution to a differential equation. Solving an IVP involves finding the general solution and then applying the initial condition to determine the constant of integration.

Integration Techniques for Exponential Functions

Solving differential equations involving exponential functions often requires integrating expressions like e^(t*y). Recognizing when to use substitution or other integration methods is essential to handle these integrals and find explicit solutions.