# Particular solution to nonhomogeneneous 2nd order ODE

How can I find a particular solution to this equation?

$$f''(x)+x^{-1}f'(x)-f(x)\left(1+x{}^{-2}\right)-Ax^{-2}=0$$

I know that the solution of the homogeneous equation (i.e. without the last term) is of the form

$$f(x)=c_{1}K_{1}(x)+d_{1}I_{1}(x)$$

but can't work out how to construct a particular solution. Variation of parameters? Thanks!

• Variation of parameters should work. You might try some identity involving Bessel functions, orthogonality and $x^{-2}$. Commented Jan 11, 2016 at 15:30
• @Pragabhava Could you expand on that please? Commented Jan 11, 2016 at 15:33
• @Pragabhava They can be imposed at the end (at least modulo singularities).
– Ian
Commented Jan 11, 2016 at 15:37
• OK, but I have to warn you that in my experience as student and teacher, it's a very undidactic way to learn (and work with) odes and pdes. A differential equation is not properly defined without boundary conditions and a domain of integration. Commented Jan 11, 2016 at 15:45
• @Pragabhava A linear ODE on a fixed domain can be usefully defined without requiring boundary conditions. The solutions form an affine space, which is the reason we seek particular solutions and homogeneous solutions separately when we attempt to identify general solutions. When it is possible to identify general solutions, this is a good technique for solving IVPs/BVPs, since adjoining the initial or boundary conditions then amounts to solving a system of linear equations. Specific boundary conditions are more important in PDE for many reasons, but the problem at hand here is an ODE.
– Ian
Commented Jan 11, 2016 at 16:24

Let's say that you have initial conditions at $x = a$ and your domain of integration is $x\in(a,b)$. The variation of parameters formula says that: $$f_p(x) = - A I_1(x)\int_a^x \frac{K_1(t) dt}{W(t) t^2} + A K_1(x)\int_a^x \frac{I_1(t)dt}{W(t)t^2},$$ where $W(t)$ is thhe Wronskian of $K_1(t)$ and $I_1(t)$, which is1 $$W(t) = K_1(t)I_1'(t) - K_1'(t) I_1(t) = \frac{1}{t}.$$
And so $$f_p(x) = -A K_1(x)\int_a^x \frac{I_1(t) dt}{t} + A I_1(x)\int_a^x \frac{K_1(t)dt}{t}.$$