15.14.9 problem 9

Internal problem ID [3181]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 23, page 106
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 04:05:03 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 40

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

\[ y = c_2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {{\mathrm e}^{x} \left (x +3 c_{1} \right )}{3} \]

✓ Mathematica. Time used: 0.4 (sec). Leaf size: 62

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

\[ y(x)\to \frac {1}{3} e^{-x/2} \left (e^{3 x/2} (x-1+3 c_1)+3 c_2 \cos \left (\frac {\sqrt {3} x}{2}\right )+3 c_3 \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.143 (sec). Leaf size: 39

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

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