55.30.7 problem 116

Internal problem ID [13889]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 116
Date solved : Thursday, October 02, 2025 at 08:08:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 31

ode:=x^2*diff(diff(y(x),x),x)-(a*x^3+5/16)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \sinh \left (\frac {2 x^{{3}/{2}} \sqrt {a}}{3}\right )+c_2 \cosh \left (\frac {2 x^{{3}/{2}} \sqrt {a}}{3}\right )}{x^{{1}/{4}}} \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 60

ode=x^2*D[y[x],{x,2}]-(a*x^3+5/16)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-\frac {2}{3} \sqrt {a} x^{3/2}} \left (2 c_1 e^{\frac {4}{3} \sqrt {a} x^{3/2}}-\frac {c_2}{\sqrt {a}}\right )}{2 \sqrt [4]{x}} \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - (a*x**3 + 5/16)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} J_{\frac {1}{2}}\left (\frac {2 x^{\frac {3}{2}} \sqrt {- a}}{3}\right ) + C_{2} Y_{\frac {1}{2}}\left (\frac {2 x^{\frac {3}{2}} \sqrt {- a}}{3}\right )\right ) \]