88.19.8 problem 8

Internal problem ID [24136]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 139
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:00:08 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 61

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

\[ y = \frac {\left (-54 x^{3}-54 x^{2}+\left (972 c_3 -36\right ) x +972 c_1 -12\right ) {\mathrm e}^{-x}}{972}+\frac {\left (36 x^{3}-72 x^{2}+972 c_2 +72 x -32\right ) {\mathrm e}^{2 x}}{972}-\frac {x}{2}-\frac {1}{4} \]

✓ Mathematica. Time used: 0.514 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to \frac {1}{972} e^{-x} \left (-6 \left (9 x^3+9 x^2+(6-162 c_2) x+2-162 c_1\right )+4 e^{3 x} \left (9 x^3-18 x^2+18 x-8+243 c_3\right )-243 e^x (2 x+1)\right ) \end{align*}

✓ Sympy. Time used: 0.293 (sec). Leaf size: 49

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

\[ y{\left (x \right )} = - \frac {x}{2} + \left (C_{1} + x \left (C_{2} - \frac {x^{2}}{18} - \frac {x}{18}\right )\right ) e^{- x} + \left (C_{3} + \frac {x^{3}}{27} - \frac {2 x^{2}}{27} + \frac {2 x}{27}\right ) e^{2 x} - \frac {1}{4} \]