60.3.159 problem 1163

Internal problem ID [11169]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1163
Date solved : Monday, January 27, 2025 at 10:47:37 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (-v^{2}+x^{2}\right ) y-f \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (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)
 

\[ y = \operatorname {BesselJ}\left (v , x\right ) c_{2} +\operatorname {BesselY}\left (v , x\right ) c_{1} -\frac {\pi \left (\int \frac {\operatorname {BesselY}\left (v , x\right ) f}{x}d x \right ) \operatorname {BesselJ}\left (v , x\right )}{2}+\frac {\pi \left (\int \frac {\operatorname {BesselJ}\left (v , x\right ) f}{x}d x \right ) \operatorname {BesselY}\left (v , x\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.159 (sec). Leaf size: 72

DSolve[-f[x] + (-v^2 + x^2)*y[x] + x*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \operatorname {BesselJ}(v,x) \int _1^x-\frac {\pi \operatorname {BesselY}(v,K[1]) f(K[1])}{2 K[1]}dK[1]+\operatorname {BesselY}(v,x) \int _1^x\frac {\pi \operatorname {BesselJ}(v,K[2]) f(K[2])}{2 K[2]}dK[2]+c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x) \]