When more than one particular solution fits the initial conditions
So I've got a general solution to a differential equation:
y = x/4 (ln x + D)^2
and I'm to solve it for the general initial value problem: y(x0) = y0, x>0, y>0
I then solved the equation for D, getting two results:
D = -ln (x0) + 2(y0/x0)^(1/2)
D = -ln (x0) - 2(y0/x0)^(1/2)
My problem now is that I have no idea how to pick one solution for D over the other. Both fit all restrictions as far as I can tell. Is it possible that both are valid and that I would need further restrictions to choose one of them?
Re: When more than one particular solution fits the initial conditions
Both solutions are correct, meaning that with either value of D the equation meets the condition that y(x_0) = y_0. I have attached a plot showing the two solutions for (X_0,Y_0) = (1,1) - note that they both pass through (1,1). You can't pick one over the other without introducing some other constraint, such as the slope at x_0 or the value for y at another point.