43.10.2 problem 1(b)

Internal problem ID [8946]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 89
Problem number : 1(b)
Date solved : Tuesday, September 30, 2025 at 06:00:24 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-8 y&={\mathrm e}^{i x} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 41

ode:=diff(diff(diff(y(x),x),x),x)-8*y(x) = exp(I*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (-\frac {8}{65}+\frac {i}{65}\right ) {\mathrm e}^{i x}+c_1 \,{\mathrm e}^{2 x}+c_2 \,{\mathrm e}^{-x} \cos \left (\sqrt {3}\, x \right )+c_3 \,{\mathrm e}^{-x} \sin \left (\sqrt {3}\, x \right ) \]

✓ Mathematica. Time used: 0.35 (sec). Leaf size: 181

ode=D[y[x],{x,3}]-8*y[x]==Exp[I*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{60} e^{-x} \left (60 \cos \left (\sqrt {3} x\right ) \int _1^x-\frac {e^{(1+i) K[1]} \left (\sqrt {3} \cos \left (\sqrt {3} K[1]\right )-3 \sin \left (\sqrt {3} K[1]\right )\right )}{12 \sqrt {3}}dK[1]+60 \sin \left (\sqrt {3} x\right ) \int _1^x-\frac {e^{(1+i) K[2]} \left (3 \cos \left (\sqrt {3} K[2]\right )+\sqrt {3} \sin \left (\sqrt {3} K[2]\right )\right )}{12 \sqrt {3}}dK[2]-(2+i) e^{(1+i) x}+60 c_1 e^{3 x}+60 c_2 \cos \left (\sqrt {3} x\right )+60 c_3 \sin \left (\sqrt {3} x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*y(x) - exp(x*complex(0, 1)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{3} e^{2 x} + \left (C_{1} \sin {\left (\sqrt {3} x \right )} + C_{2} \cos {\left (\sqrt {3} x \right )}\right ) e^{- x} + \frac {e^{x \operatorname {complex}{\left (0,1 \right )}}}{\operatorname {complex}^{3}{\left (0,1 \right )} - 8} \]