Internal problem ID [8760]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+3 x y^{\prime }+\left (-v^{2}+x^{2}+1\right ) y-f \relax (x )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \relax (x ) = \frac {\BesselJ \left (v , x\right ) c_{2}}{x}+\frac {\BesselY \left (v , x\right ) c_{1}}{x}+\frac {\pi \left (\BesselY \left (v , x\right ) \left (\int \BesselJ \left (v , x\right ) f \relax (x )d x \right )-\BesselJ \left (v , x\right ) \left (\int \BesselY \left (v , x\right ) f \relax (x )d x \right )\right )}{2 x} \]
✓ Solution by Mathematica
Time used: 0.048 (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 {J_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 J_v(K[2]) f(K[2])dK[2]+c_2\right )}{x} \\ \end{align*}