81.11.27 problem 15-26

Internal problem ID [21651]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 15. Method of undetermined coefficients. Page 337.
Problem number : 15-26
Date solved : Thursday, October 02, 2025 at 07:59:28 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=2 x \,{\mathrm e}^{-x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 32

ode:=diff(diff(diff(y(x),x),x),x)-6*diff(diff(y(x),x),x)+11*diff(y(x),x)-6*y(x) = 2*x*exp(-x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-12 x -13\right ) {\mathrm e}^{-x}}{144}+c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{2 x}+c_3 \,{\mathrm e}^{3 x} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

ode=D[y[x],{x,3}]-6*D[y[x],{x,2}]+11*D[y[x],x]-6*y[x]==2*x*Exp[-x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{144} e^{-x} (12 x+13)+c_1 e^x+c_2 e^{2 x}+c_3 e^{3 x} \end{align*}

✓ Sympy. Time used: 0.176 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*exp(-x) - 6*y(x) + 11*Derivative(y(x), x) - 6*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x} + \frac {\left (- 12 x - 13\right ) e^{- x}}{144} \]