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)

In Exercises 23–28, solve the initial value problem.

x dy/dx + 2y = x² + 1, x > 0, y(1) = 1

In Exercises 23–28, solve the initial value problem.

x dy + (y - cos x) dx = 0, y(π/2) = 0

133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

1. 2y' + 3y = e^(-x)

a. y = e^(-x)

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x

In Exercises 5–8, show that each function is a solution of the given initial value problem.

5. Differential Equation: 2y + y' = 4x + 2

Initial condition: y(-1) = e² - 2

Solution candidate: y = e^(-2x) + 2x

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ∫₁ ͯ 1/t dt

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = 1 + ∫₀ ͯ y(t) dt