Good day everybody. I'm a physics student enrolled in a scientific computation course and my teacher sent us a bunch of ODEs to solve both analitically and numerically. I haven't really studied boundary value problems so maybe I need to be pointed to a resource for that but right now I'm but having trouble with this one in particular: $$u'' + 9u = 36(5- |x|)^2$$ with the boundary conditions $u(-5) = 0$ and $u(5) = 0$.
My attempt at solving it goes like this:
- I expand and separate the equation in its two domains $$u"+ 9u = 36x^2 + 360x + 900, \ x < 0$$ $$u"+ 9u = 36x^2 - 360x + 900, \ x \ge 0$$ Now I have two equations so I believe I need two homogeneous solutions, two particular solutions and four conditions. Therefore
- I solve for to the homogeneous part: $$U_h(x) = A\cos(3x) + B\sin(3x), \ x<0$$ $$U_h(x) = C\cos(3x) + D\sin(3x), \ x\ge 0$$
- I resort to undetermined coefficients looking for particular solutions: $$U_p(x) = 4x^2+40x + 892/9, \ x < 0$$ $$U_p(x) = 4x^2-40x + 892/9, \ x \ge 0$$
- I add $U_h$ and $U_p$ to make two general solutions: $$U(x) = A\cos(3x) + B\sin(3x) + 4x^2+40x + \frac{892}{9}, \ x<0 $$ $$U(x) = C\cos(3x) + D\sin(3x) + 4x^2-40x + \frac{892}{9}, \ x\ge0 $$
- This is where I notice I have only two boundary conditions, not four.
So, there is that. Should I merge both solutions back togheter into one? or maybe two conditions are enough and that's what I'm not seeing. Anyway, I hope you guys can help.