60.3.215 problem 1219

Internal problem ID [11225]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1219
Date solved : Tuesday, January 28, 2025 at 05:42:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.138 (sec). Leaf size: 65

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

\[ y = {\mathrm e}^{-\int \frac {f}{x}d x} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.266 (sec). Leaf size: 151

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

\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i b}{\sqrt {a}}+\sqrt {1-4 c}+1\right ),\sqrt {1-4 c}+1,2 i \sqrt {a} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i b}{\sqrt {a}}-\sqrt {1-4 c}-1\right )}^{\sqrt {1-4 c}}\left (2 i \sqrt {a} x\right )\right ) \exp \left (\int _1^x\frac {-2 f(K[1])-2 i \sqrt {a} K[1]+\sqrt {1-4 c}+1}{2 K[1]}dK[1]\right ) \]