Internal problem ID [8743]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1163.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y-f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
dsolve(x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(-v^2+x^2)*y(x)-f(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \BesselJ \left (v , x\right ) c_{2}+\BesselY \left (v , x\right ) c_{1}-\frac {\pi \left (\BesselJ \left (v , x\right ) \left (\int \frac {\BesselY \left (v , x\right ) f \relax (x )}{x}d x \right )-\BesselY \left (v , x\right ) \left (\int \frac {\BesselJ \left (v , x\right ) f \relax (x )}{x}d x \right )\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.078 (sec). Leaf size: 66
DSolve[-f[x] + (-v^2 + x^2)*y[x] + x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to J_v(x) \left (\int _1^x-\frac {\pi Y_v(K[1]) f(K[1])}{2 K[1]}dK[1]+c_1\right )+Y_v(x) \left (\int _1^x\frac {\pi J_v(K[2]) f(K[2])}{2 K[2]}dK[2]+c_2\right ) \\ \end{align*}