Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.


t dy/dt + 2y = t³, t > 0, y(2) = 1

Integral Equations

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

and initial condition for y.

y = 1 + ∫₀ ͯ y(t) dt

Using Euler’s Method

In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.

y' = 2xy + 2y, y(0) = 3, dx = 0.2

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.

x²y' + 2xy = y³

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)?