67.16.12 problem 24.1 (L)

Internal problem ID [16811]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 24. Variation of parameters. Additional exercises page 444
Problem number : 24.1 (L)
Date solved : Thursday, October 02, 2025 at 01:39:16 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 26

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

\[ y = \sinh \left (x^{2}\right ) c_2 +\cosh \left (x^{2}\right ) c_1 +\frac {x^{2} {\mathrm e}^{x^{2}}}{8} \]

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 47

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

\begin{align*} y(x)&\to \frac {1}{16} \left (\left (2 x^2-1+16 c_1\right ) \cosh \left (x^2\right )+\sinh \left (x^2\right ) \left (\log \left (e^{2 x^2}\right )-1+16 i c_2\right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x*(-4*x**2*y(x) - x**2*exp(x**2) + Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method