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).

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

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

(x+1) dy/dx - 2 (x² + x)y = exp(x²) / (x+1), x > -1, y(0) = 5

Solving Initial Value Problems

Solve the initial value problems in Exercises 15–20.

t dy/dt + 2y = t³, t > 0, y(2) = 1

Solve the Bernoulli equations in Exercises 29–32.

x²y' + 2xy = y³

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)?

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

xdy/dx + y = e ͯ, x > 0