12.9.16 problem 16

Internal problem ID [1772]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 16
Date solved : Saturday, March 29, 2025 at 11:39:01 PM
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&=4 x^{{5}/{2}} {\mathrm e}^{2 x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\sqrt {x} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 31

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

\[ y(x)\to \frac {1}{2} \sqrt {x} \left (e^{2 x}+2 c_2 e^x+2 c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**(5/2)*exp(2*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(2*x) + x**2*Derivative(y(x), (x, 2)) + x*y(x)/2 + 3*y(x)/4)/(x*(x + 1)) cannot be solved by the factorable group method