78.16.3 problem 3

Internal problem ID [18311]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 23. Operator Methods for Finding Particular Solutions. Problems at page 169
Problem number : 3
Date solved : Monday, March 31, 2025 at 05:25:19 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y^{\prime }+4 y&=10 x^{3} {\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+4*diff(y(x),x)+4*y(x) = 10*x^3*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-2 x} \left (c_2 +c_1 x +\frac {1}{2} x^{5}\right ) \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 27

ode=D[y[x],{x,2}]+4*D[y[x],x]+4*y[x]==10*x^3*Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.315 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-10*x**3*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)
 

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