5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.
e⁴ᵗy'(t) = 5

Verified step by step guidance

1. Identify the given differential equation: \(e^{4t} y'(t) = 5\).
2. Rewrite the equation to isolate \(y'(t)\): \(y'(t) = \frac{5}{e^{4t}} = 5 e^{-4t}\).
3. Recognize that this is a separable differential equation where \(y'(t)\) is expressed explicitly in terms of \(t\).
4. Integrate both sides with respect to \(t\) to find \(y(t)\): \(y(t) = \int 5 e^{-4t} \, dt\).
5. Perform the integration using the formula \(\int e^{at} dt = \frac{1}{a} e^{at} + C\), and include the constant of integration \(C\) to express the general solution.

Key Concepts

Separable Differential Equations

A separable differential equation can be written so that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other. This allows integration of both sides separately to find the general solution.

Integration of Both Sides

After separating variables, integrate each side with respect to its variable. This step is crucial to find an implicit or explicit form of the solution, often involving an integration constant representing the general solution.

Expressing the Solution Explicitly

Once integrated, solve the resulting equation for the dependent variable explicitly as a function of the independent variable. This often involves algebraic manipulation to isolate the dependent variable on one side.