Explain how to solve a separable differential equation of the form
g(t)y'(y) = h(t)

Verified step by step guidance

1. Identify the given differential equation: \( g(t) y'(y) = h(t) \). Note that \( y' \) typically denotes \( \frac{dy}{dt} \), so clarify the variables involved and the form of the equation.
2. Rewrite the equation to isolate \( \frac{dy}{dt} \). Since \( y' = \frac{dy}{dt} \), express the equation as \( g(t) \frac{dy}{dt} = h(t) \).
3. Separate the variables by rewriting the equation so that all terms involving \( y \) are on one side and all terms involving \( t \) are on the other. This might involve dividing both sides by \( g(t) \) and expressing \( \frac{dy}{dt} \) explicitly.
4. Integrate both sides with respect to their respective variables: integrate the \( y \)-side with respect to \( y \) and the \( t \)-side with respect to \( t \). This step uses the principle that \( \int dy = \int \frac{h(t)}{g(t)} dt \).
5. After integration, include the constant of integration \( C \) and solve for \( y \) explicitly if possible, to find the general solution to the differential equation.

Key Concepts

Separable Differential Equations

A separable differential equation can be written so that all terms involving one variable and its derivative are on one side, and all terms involving the other variable are on the opposite side. This allows the equation to be integrated separately with respect to each variable, simplifying the solution process.

Separation of Variables Technique

This technique involves algebraically manipulating the differential equation to isolate y and dy on one side and t and dt on the other. Once separated, both sides are integrated independently, leading to an implicit or explicit solution for y in terms of t.

Integration of Both Sides

After separating variables, integrating each side with respect to its variable is essential. This step often involves finding antiderivatives of functions g(t) and h(t), and may include adding an integration constant, which represents the family of solutions to the differential equation.