67.9.18 problem 14.2 (h)

Internal problem ID [16566]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (h)
Date solved : Thursday, October 02, 2025 at 01:36:30 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} y^{\prime \prime }-\frac {y^{\prime }}{x}-4 x^{2} y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x^{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=diff(diff(y(x),x),x)-diff(y(x),x)/x-4*x^2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sinh \left (x^{2}\right )+c_2 \cosh \left (x^{2}\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 23

ode=D[y[x],{x,2}]-1/x*D[y[x],x]-4*x^2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cosh \left (x^2\right )+i c_2 \sinh \left (x^2\right ) \end{align*}

✓ Sympy. Time used: 0.111 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2*y(x) + Derivative(y(x), (x, 2)) - Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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