Problem 2: Solve the initial value problem y' = xy y(0) = 1 ​

Question

Problem 2: Solve the initial value problem

`y' = xy`

`y(0) = 1`
​

Answer 1

Separate the variables:

y' = dy/dx = xy ⇒ 1/y dy = x dx

Integrate both sides:

∫ 1/y dy = ∫ x dx

ln|y| = 1/2 x² + C

Given that y(0) = 1, we have

ln|1| = 1/2 • 0² + C ⇒ C = 0

so that the particular solution is

ln|y| = 1/2 x²

Solving for y gives

y = exp(1/2 x²)