First-Order Linear Equations
Solve the differential equations in Exercises 1–14.


xdy/dx + y = e ͯ, x > 0

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

(t-1)³ ds/dt + 4(t-1)²s = t+1, t >1

Show that the solution of the initial value problem

y' = x + y, y(x₀) = y₀

is

y = -1 -x + (1 + x₀ + y₀) exp(x-x₀).

Use Euler’s method with dx = 0.5 to estimate y(5) if y′ = y²/√x and y(1) = −1. What is the exact value of y(5)?

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

dy/dt + 2y = 3, y(0) = 1

Solve the Bernoulli equations in Exercises 29–32.

y' - y = xy²

What integral equation is equivalent to the initial value problem y' = f(x), y(x₀) = y₀?