In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)x)

Verified step by step guidance

1. Identify the given differential equation: \(2y' + 3y = e^{-x}\) and the proposed solution: \(y = e^{-x} + Ce^{-\frac{3}{2}x}\).
2. Compute the derivative \(y'\) of the given function \(y\). Since \(y = e^{-x} + Ce^{-\frac{3}{2}x}\), use the chain rule to find \(y' = -e^{-x} - \frac{3}{2}Ce^{-\frac{3}{2}x}\).
3. Substitute \(y\) and \(y'\) into the left-hand side of the differential equation: calculate \(2y' + 3y = 2\left(-e^{-x} - \frac{3}{2}Ce^{-\frac{3}{2}x}\right) + 3\left(e^{-x} + Ce^{-\frac{3}{2}x}\right)\).
4. Simplify the expression by distributing and combining like terms carefully, paying attention to coefficients and exponents.
5. Verify that after simplification, the expression equals the right-hand side of the differential equation, \(e^{-x}\), confirming that the given function is indeed a solution.

Key Concepts

Differential Equations

A differential equation relates a function with its derivatives. Solving it means finding all functions that satisfy this relationship. In this problem, verifying a solution involves substituting the given function into the equation to check if it holds true.

Derivative of Exponential Functions

The derivative of an exponential function e^(kx) is ke^(kx). Understanding this rule is essential to compute y' when y involves exponential terms, which is necessary to substitute into the differential equation.

General Solution and Constant of Integration

The general solution to a linear differential equation often includes an arbitrary constant C, representing infinitely many solutions. Recognizing how this constant appears and affects the solution helps verify that the given function satisfies the equation for all values of C.