6.10.9 problem 9

Internal problem ID [1813]
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 : 9
Date solved : Tuesday, September 30, 2025 at 05:19:49 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y&=4 \sqrt {x}\, {\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 21

ode:=4*x^2*diff(diff(y(x),x),x)+(-8*x^2+4*x)*diff(y(x),x)+(4*x^2-4*x-1)*y(x) = 4*x^(1/2)*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (x \ln \left (x \right )+\left (c_1 -1\right ) x +c_2 \right ) {\mathrm e}^{x}}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 27

ode=4*x^2*D[y[x],{x,2}]+(4*x-8*x^2)*D[y[x],x]+(4*x^2-4*x-1)*y[x]==4*x^(1/2)*Exp[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*sqrt(x)*exp(x) + 4*x**2*Derivative(y(x), (x, 2)) + (-8*x**2 + 4*x)*Derivative(y(x), x) + (4*x**2 - 4*x - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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