6.10.22 problem 22

Internal problem ID [1826]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 05:20:08 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }-\left (x^{2}-2\right ) y&=3 x^{4} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 20

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

\[ y = x \sinh \left (x \right ) c_2 +x \cosh \left (x \right ) c_1 -3 x^{2} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to \frac {1}{2} x \left (-6 x+2 c_1 e^{-x}+c_2 e^x\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (x**2*(-3*x**2 - y(x) + Derivative(y(x), (x, 2)))/2 + y(x))/x cannot be solved by the factorable group method