5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.
y'(t) = eʸᐟ²sin t

Verified step by step guidance

1. Rewrite the given differential equation in terms of \( y \) and \( t \): \( \frac{dy}{dt} = e^{y/2} \sin t \).
2. Separate the variables by bringing all terms involving \( y \) to one side and all terms involving \( t \) to the other side: \( e^{-y/2} dy = \sin t \, dt \).
3. Integrate both sides: \( \int e^{-y/2} \, dy = \int \sin t \, dt \).
4. Evaluate the integrals: for the left side, use substitution to integrate \( e^{-y/2} \), and for the right side, recall that \( \int \sin t \, dt = -\cos t + C \).
5. After integration, solve the resulting equation explicitly for \( y \) as a function of \( t \), including the constant of integration.

Key Concepts

Separable Differential Equations

A separable differential equation can be written as a product of a function of the dependent variable and a function of the independent variable. This allows the variables to be separated on opposite sides of the equation, enabling integration with respect to each variable independently.

Integration Techniques

Solving separable equations requires integrating both sides after separation. Familiarity with integrating exponential functions and trigonometric functions, such as sin(t), is essential to find the antiderivatives and express the solution explicitly.

Implicit and Explicit Solutions

After integration, solutions may be implicit or explicit. An explicit solution expresses the dependent variable directly as a function of the independent variable, which often involves algebraic manipulation or applying inverse functions to isolate the dependent variable.