54.3.32 problem 1037

Internal problem ID [12327]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1037
Date solved : Friday, October 03, 2025 at 03:18:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 64

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

\[ y = {\mathrm e}^{-\frac {x \left (b x +a \right )}{2}} x \left (\operatorname {KummerU}\left (\frac {a^{2}+12 b +4 c}{16 b}, \frac {3}{2}, x^{2} b \right ) c_2 +\operatorname {KummerM}\left (\frac {a^{2}+12 b +4 c}{16 b}, \frac {3}{2}, x^{2} b \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 74

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

\begin{align*} y(x)&\to e^{-\frac {1}{2} x (a+b x)} \left (c_1 \operatorname {HermiteH}\left (-\frac {a^2+4 (b+c)}{8 b},\sqrt {b} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^2+4 (b+c)}{16 b},\frac {1}{2},b x^2\right )\right ) \end{align*}

✗ Sympy

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

False