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


y' + (tanx)y = cos²x, -π/2 < x < π/2

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

Integral Equations

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

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

First-Order Linear Equations

Solve the differential equations in Exercises 1–14.

(1+x)y' + y = √x

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

Solve the Bernoulli equations in Exercises 29–32.

y' - y = xy²

Solve the Bernoulli equations in Exercises 29–32.

x²y' + 2xy = y³