Solve: y′ = 10xy − 2x, y(0)=4 y(x)=?

Answer 1

It takes only a tiny bit of algebra to make this equation separable

dy/dx = 10xy - 2x

dy/dx = 2x(5y - 1)

dy/(5y - 1) = 2xdx

∫dy/(5y - 1) = ∫2xdx

ln(5y - 1)/5 = x² + C

Calculate C by substituting in the initial condition y(0)) = 4

ln(5×4 - 1)/5 = 0² + C

C = ln(19)/5

From the resulting equation express y in terms of x

ln(5y - 1)/5 = x² + ln(19)/5

ln(5y - 1) = 5x² + ln(19)

5y - 1 = e^(5x² + ln(19))

y = (19e^5x² + 1)/5