8.11.33 problem 56

Internal problem ID [901]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 56
Date solved : Tuesday, March 04, 2025 at 12:00:14 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 34

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

\[ y = -\frac {\left (-9 c_2 \,{\mathrm e}^{4 x}+3 x \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{3 x}-9 c_1 \right ) {\mathrm e}^{-2 x}}{9} \]

✓ Mathematica. Time used: 0.016 (sec). Leaf size: 34

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

\[ y(x)\to -\frac {1}{9} e^x (3 x+2)+c_1 e^{2 x}+c_2 e^{-2 x} \]

✓ Sympy. Time used: 0.103 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} e^{2 x} - \frac {x e^{x}}{3} - \frac {2 e^{x}}{9} \]