6.10.29 problem 29

Internal problem ID [1833]
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 : 29
Date solved : Tuesday, September 30, 2025 at 05:20:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x \left (x +1\right ) y^{\prime }+\left (2 x +3\right ) y&=x^{{5}/{2}} {\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 21

ode:=4*x^2*diff(diff(y(x),x),x)-4*x*(1+x)*diff(y(x),x)+(2*x+3)*y(x) = x^(5/2)*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (x +4 c_1 \right ) {\mathrm e}^{x}+4 c_2 \right ) \sqrt {x}}{4} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 30

ode=4*x^2*D[y[x],{x,2}]-4*x*(x+1)*D[y[x],x]+(2*x+3)*y[x]==x^(5/2)*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \sqrt {x} \left (e^x (x-1+4 c_2)+4 c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**(5/2)*exp(x) + 4*x**2*Derivative(y(x), (x, 2)) - 4*x*(x + 1)*Derivative(y(x), x) + (2*x + 3)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**(5/2)*exp(x) + 4*x**2*Derivative(y(x), (x, 2)) + 2*x*y(x) + 3*y(x))/(4*x*(x + 1)) cannot be solved by the factorable group method