20.11.8 problem Problem 11

Internal problem ID [3790]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number : Problem 11
Date solved : Tuesday, March 04, 2025 at 05:16:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 25

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

\[ y \left (x \right ) = 2 x^{2} {\mathrm e}^{2 x}+c_{1} {\mathrm e}^{2 x}+2 c_{2} x +c_{2} \]

✓ Mathematica. Time used: 0.043 (sec). Leaf size: 32

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

\[ y(x)\to e^{2 x} \left (2 x^2-1+c_1\right )-\frac {1}{4} c_2 (2 x+1) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-8*x**2*exp(2*x) + x*Derivative(y(x), (x, 2)) + 2*y(x))/(2*x + 1) cannot be solved by the factorable group method