60.3.216 problem 1220

Internal problem ID [11226]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1220
Date solved : Tuesday, January 28, 2025 at 05:42:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 38

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

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

✓ Solution by Mathematica

Time used: 0.366 (sec). Leaf size: 62

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

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