How do I find the exact solution to the boundary value problem $y''(x) = −xy' (x) + y(x) − 1$, $y(0) = 1$, $y(4) = 9$?

Question

I believe I need to guess a general solution of $y$ for this boundary value problem. However I am not sure what that guess is. Can someone explain if I am on the right track or not? If so what is the guess for the general solution of y?

Answer 1

We try $y = ax+b$. Substitute into the original equation, we get $b=1$. It happens to fit the first boundary condition. The second boundary condition gives $a=2$. Since the boundary condition uniquely determines this equation, this above solution is the unique one.

For a more general setting $y''(x) = −xy' (x) + y(x) − \beta$, $\beta\ne 1$, the above trial solution ceases to work. But after differentiating the equation once, we have $$y'''=-xy''.$$ It is a first order ODE of $y''$. Solve, $$\frac{y''(x)}{y''(x_0)}=\exp\bigg(-\frac{x^2-x_0^2}{2}\bigg),$$ for some $x_0$ where $y''(x_0)\ne 0$. $x_0$ exists. Otherwise $y$ is reduced to the linear form that fails to satisfy the equation.

You can integrate and match the boundary condition to arrive at the solution.