I have this diffyQ problem that asks to find the solution that passes through a specific point. I have worked it out to where I have a ( i think correct ) equation for y, but am not sure how to deal with the constant and make sure it passes through a specific point.
Solve: Find the solution of y' = 2(2x-y) that passes through the point P(0,1).
y' = 4x -2y
y' + 2y = 4x
integrating factor is e^(2x)
d/dx of [e^(2x)y] = 4xe^(2x)
integrate both sides
RHS by parts gives
2xe^(2x) - e^2x
e^(2x)y = 2xe^(2x) - e^(2x) + C
y = 2x - 1 + Ce^(-2x)
Could someone check this out and tell me how to get a solution that passes through that point? Thanks!