Internal problem ID [9493]
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]]
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x +\left (-v^{2}+x^{2}\right ) y=f \left (x \right )} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 49
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)
\[ y \left (x \right ) = \operatorname {BesselJ}\left (v , x\right ) c_{2} +\operatorname {BesselY}\left (v , x\right ) c_{1} +\frac {\pi \left (\int \frac {\operatorname {BesselJ}\left (v , x\right ) f \left (x \right )}{x}d x \right ) \operatorname {BesselY}\left (v , x\right )}{2}-\frac {\pi \left (\int \frac {\operatorname {BesselY}\left (v , x\right ) f \left (x \right )}{x}d x \right ) \operatorname {BesselJ}\left (v , x\right )}{2} \]
✓ Solution by Mathematica
Time used: 0.165 (sec). Leaf size: 72
DSolve[-f[x] + (-v^2 + x^2)*y[x] + x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \operatorname {BesselJ}(v,x) \int _1^x-\frac {\pi \operatorname {BesselY}(v,K[1]) f(K[1])}{2 K[1]}dK[1]+\operatorname {BesselY}(v,x) \int _1^x\frac {\pi \operatorname {BesselJ}(v,K[2]) f(K[2])}{2 K[2]}dK[2]+c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x) \]