23.3.103 problem 105

Internal problem ID [5817]
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 : 105
Date solved : Friday, October 03, 2025 at 01:43:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.046 (sec). Leaf size: 88

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

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

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 117

ode=(b + c*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 \left (-a-i \sqrt {c} x\right )} \left (c_1 \operatorname {HermiteH}\left (\frac {i \left (a^2-4 b+4 i \sqrt {c}\right )}{8 \sqrt {c}},\sqrt [4]{-1} \sqrt [4]{c} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {i \left (a^2-4 b+4 i \sqrt {c}\right )}{16 \sqrt {c}},\frac {1}{2},i \sqrt {c} 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 + c*x**2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False