89.18.23 problem 23

Internal problem ID [24723]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 9. Nonhomogeneous Equations: Undetermined coefficients. Oral Exercises at page 146
Problem number : 23
Date solved : Thursday, October 02, 2025 at 10:47:27 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

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

\[ y = \frac {{\mathrm e}^{2 x}}{7}+c_1 \,{\mathrm e}^{x}+c_2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \]

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 63

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

\begin{align*} y(x)&\to \frac {1}{7} e^{-x/2} \left (e^{3 x/2} \left (e^x+7 c_1\right )+7 c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+7 c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 42

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

\[ y{\left (x \right )} = C_{3} e^{x} + \left (C_{1} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} + \frac {e^{2 x}}{7} \]