21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = 3 + e⁻²ᵗ

Verified step by step guidance

1. Recognize that the given differential equation is a first-order ordinary differential equation of the form \(y'(t) = f(t)\), where \(f(t) = 3 + e^{-2t}\).
2. To find the general solution, integrate both sides with respect to \(t\): \(\int y'(t) \, dt = \int (3 + e^{-2t}) \, dt\).
3. Split the integral on the right side into two separate integrals: \(\int 3 \, dt + \int e^{-2t} \, dt\).
4. Integrate each term: the integral of \(3\) with respect to \(t\) is \$3t\(, and the integral of \)e^{-2t}\( with respect to \)t$ is \(-\frac{1}{2} e^{-2t}\) (using substitution or recognizing the exponential integral).
5. Combine the results and add the arbitrary constant of integration \(C\) to write the general solution: \(y(t) = 3t - \frac{1}{2} e^{-2t} + C\).

Key Concepts

General Solution of a Differential Equation

The general solution of a differential equation includes all possible solutions and typically contains arbitrary constants. It represents the family of functions that satisfy the equation, capturing both particular and homogeneous parts.

Integration of Functions

Solving first-order differential equations often involves integrating the right-hand side with respect to the independent variable. Integration reverses differentiation and helps find the original function from its derivative.

Exponential Functions and Their Integrals

Exponential functions like e^(-2t) have specific integral formulas. For example, ∫e^(kt) dt = (1/k)e^(kt) + C, where k is a constant. Recognizing and integrating these correctly is essential for solving differential equations involving exponentials.