60.3.217 problem 1221

Internal problem ID [11227]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1221
Date solved : Tuesday, January 28, 2025 at 05:42:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 35

dsolve(x^2*diff(diff(y(x),x),x)+(x-2*x^2*f(x))*diff(y(x),x)+(x^2*(1+f(x)^2-diff(f(x),x))-x*f(x)-v^2)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.219 (sec). Leaf size: 31

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

\[ y(x)\to (c_1 \operatorname {BesselJ}(v,x)+c_2 \operatorname {BesselY}(v,x)) \exp \left (\int _1^xf(K[1])dK[1]\right ) \]