87.14.4 problem 4

Internal problem ID [23523]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 2. Linear differential equations. Exercise at page 109
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:42:42 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{4} y^{\prime \prime }+2 x^{3} y^{\prime }-y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{\frac {1}{x}} \end{align*}

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

ode:=x^4*diff(diff(y(x),x),x)+2*x^3*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sinh \left (\frac {1}{x}\right )+c_2 \cosh \left (\frac {1}{x}\right ) \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 23

ode=x^4*D[y[x],{x,2}]+(2*x^3)*D[y[x],x]-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cosh \left (\frac {1}{x}\right )-i c_2 \sinh \left (\frac {1}{x}\right ) \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + 2*x**3*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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