12.9.14 problem 14

Internal problem ID [1770]
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 : 14
Date solved : Saturday, March 29, 2025 at 11:38:59 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 19

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

\[ y = {\mathrm e}^{-x} \left (x^{{3}/{2}}+\sqrt {x}\, c_1 +c_2 \right ) \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 28

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

\[ y(x)\to e^{-x} \left (x^{3/2}+2 c_2 \sqrt {x}+c_1\right ) \]

✗ Sympy

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

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