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)?

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

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)?

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

(x²+y²)dx + xy dy = 0

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