Nonhomogeneous second order ODE constant external force I've been trying to solve this equation $$-\mu u'' + \beta u' = 1$$
where $u(0) = 0$, $u'(1) = 1$. So far the result I have is $$u = \frac{(exp(\frac{\beta x}{\mu})-1)\mu}{\beta exp(\frac{\beta}{\mu})}$$ but when I tried to substitute it on the equation the result is not the same.
What am I missing? I know that to get the solution we have to find two "solutions" the general and the specific one, can anyone explain how to solve this?

Break the problem up into two parts: getting the general solution of the homogeneous problem as well as finding a particular solution to the nonhomogeneous problem.
Homogeneous ODE: $-\mu u''+\beta u'=0$
Use the standard methods for second order, linear, constant coefficient ODEs to get $u_h(x)=c_1+c_2e^{\beta x/\mu}$, the general solution of the homogeneous problem.
Nonhomogeneous ODE: $-\mu u''+\beta u'=1$
There are multiple methods to find a particular solution $u_p$ of the nonhomogeneous problem. I prefer the method of undetermined coefficients here, but your mileage may vary. Doing so yields $u_p(x)={1\over \beta}x$.
Then, the general solution to the nonhomgeneous problem is $u(x)=u_h(x)+u_p(x)$.
Finally, you would need to apply the two given boundary conditions to find the two constants in $u(x)$, but this is just a little calculus and algebra.
