23.3.15 problem 8(e)

Internal problem ID [4156]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 4. The general linear differential equation of order n. Exercises at page 63
Problem number : 8(e)
Date solved : Sunday, March 30, 2025 at 02:41:11 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+4 y&=x^{3} {\mathrm e}^{2 x}+x \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 28

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

\[ y = \frac {{\mathrm e}^{2 x} \left (3 x^{5}+10 x^{3}+60 c_1 x +60 c_2 \right )}{60} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 34

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

\[ y(x)\to \frac {1}{60} e^{2 x} \left (3 x^5+10 x^3+60 c_2 x+60 c_1\right ) \]

✓ Sympy. Time used: 0.318 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x^{4}}{20} + \frac {x^{2}}{6}\right )\right ) e^{2 x} \]