23.3.39 problem 39

Internal problem ID [5753]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 39
Date solved : Friday, October 03, 2025 at 01:43:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 109

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

\[ y = {\mathrm e}^{-\frac {i x \left (c x +b \right )}{2 \sqrt {c}}} \left (c_2 \left (2 c x +b \right ) \operatorname {hypergeom}\left (\left [\frac {12 c^{{3}/{2}}+4 i a c -i b^{2}}{16 c^{{3}/{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 c x +b \right )^{2}}{4 c^{{3}/{2}}}\right )+c_1 \operatorname {hypergeom}\left (\left [\frac {4 c^{{3}/{2}}+4 i a c -i b^{2}}{16 c^{{3}/{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 c x +b \right )^{2}}{4 c^{{3}/{2}}}\right )\right ) \]

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 110

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

\begin{align*} y(x)&\to c_2 \operatorname {ParabolicCylinderD}\left (\frac {-i b^2-4 c^{3/2}+4 i a c}{8 c^{3/2}},-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) (b+2 c x)}{c^{3/4}}\right )+c_1 \operatorname {ParabolicCylinderD}\left (\frac {i \left (b^2+4 i c^{3/2}-4 a c\right )}{8 c^{3/2}},\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) (b+2 c x)}{c^{3/4}}\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 + b*x + c*x**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False