Solve the differential equation y prime equals the product of 6 times x and the square root of the quantity 1 minus y squared . Explain why the initial value problem y prime equals the product...

Question

1. Solve the differential equation  .
2. Explain why the initial value problem  with y(0) = 2 does not have a solution.

Answer 1

1) ∫dy/√(1-y²) = 6∫xdx

Letting y = sin(θ), dy = cos(θ)

∫cos(θ)/cos(θ)dθ = θ

θ = 3x²+C

sin^-1(y) = 3x²+C

y = sin(3x²+C)

2) if y(0) = 2 then sin(C) =2, but the range of sin(C) = ±1, so no solution