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

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' = x(1-y), y(1) = 0, dx = 0.2

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

y' = y/x + cos ((y-x)/x)

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

Use Euler’s method with dx = 0.2 to estimate y(1) if y′ = y and y(0) = 1. What is the exact value of y(1)?

Use Euler’s method with dx = 1/3 to estimate y(2) if y′ = x sin y and y(0) = 1. What is the exact value of y(2)?

Solve the homogeneous equations in Exercises 5–10. First put the equation in the form of a homogeneous equation.

(x.exp(y/x) + y)dx - x dy = 0