54.3.176 problem 1190

Internal problem ID [12471]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1190
Date solved : Friday, October 03, 2025 at 03:19:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.040 (sec). Leaf size: 38

ode:=x^2*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+(a*x+b)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.531 (sec). Leaf size: 105

ode=(b + a*x)*y[x] + x^2*D[y[x],x] + x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

ValueError : Expected Expr or iterable but got None