60.3.204 problem 1219

Internal problem ID [11200]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1219
Date solved : Sunday, March 30, 2025 at 07:45:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 65

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\int \frac {f \left (x \right )}{x}d x} \left (c_2 \operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right )+c_1 \operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i \sqrt {a}\, x \right )\right ) \]

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

ode=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; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
f = Function("f") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + 2*x*f(x)*Derivative(y(x), x) + (a*x**2 + b*x + c + x*Derivative(f(x), x) + f(x)**2 - f(x))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None