2.11.9 problem 9

Internal problem ID [877]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:18:55 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-3 y&=1+x \,{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

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

\[ y = -\frac {1}{3}+{\mathrm e}^{-3 x} c_2 +\frac {\left (8 x^{2}+64 c_1 -4 x +1\right ) {\mathrm e}^{x}}{64} \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 38

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

\begin{align*} y(x)&\to \frac {1}{64} e^x \left (8 x^2-4 x+1+64 c_2\right )+c_1 e^{-3 x}-\frac {1}{3} \end{align*}

✓ Sympy. Time used: 0.162 (sec). Leaf size: 26

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

\[ y{\left (x \right )} = C_{2} e^{- 3 x} + \left (C_{1} + \frac {x^{2}}{8} - \frac {x}{16}\right ) e^{x} - \frac {1}{3} \]