Internal problem ID [8758]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1180.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x +\left (-v^{2}+x^{2}+1\right ) y-f \left (x \right )=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 53
dsolve(x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+(-v^2+x^2+1)*y(x)-f(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {BesselJ}\left (v , x\right ) c_{2}}{x}+\frac {\operatorname {BesselY}\left (v , x\right ) c_{1}}{x}-\frac {\pi \left (\left (\int \operatorname {BesselY}\left (v , x\right ) f \left (x \right )d x \right ) \operatorname {BesselJ}\left (v , x\right )-\left (\int \operatorname {BesselJ}\left (v , x\right ) f \left (x \right )d x \right ) \operatorname {BesselY}\left (v , x\right )\right )}{2 x} \]
✓ Solution by Mathematica
Time used: 0.044 (sec). Leaf size: 62
DSolve[-f[x] + (1 - v^2 + x^2)*y[x] + 3*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\operatorname {BesselJ}(v,x) \left (\int _1^x-\frac {1}{2} \pi Y_v(K[1]) f(K[1])dK[1]+c_1\right )+Y_v(x) \left (\int _1^x\frac {1}{2} \pi \operatorname {BesselJ}(v,K[2]) f(K[2])dK[2]+c_2\right )}{x} \\ \end{align*}