60.3.176 problem 1180

Internal problem ID [11186]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1180
Date solved : Monday, January 27, 2025 at 10:48:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (-v^{2}+x^{2}+1\right ) y-f \left (x \right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 49

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)
 

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

✓ Solution by Mathematica

Time used: 0.082 (sec). Leaf size: 68

DSolve[-f[x] + (1 - v^2 + x^2)*y[x] + 3*x*D[y[x],x] + x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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