73.17.47 problem 47

Internal problem ID [15510]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 25. Review exercises for part III. page 447
Problem number : 47
Date solved : Monday, March 31, 2025 at 01:39:59 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+8 y&={\mathrm e}^{-2 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(-2*x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.449 (sec). Leaf size: 169

ode=D[y[x],{x,3}]+8*y[x]==Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.157 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(8*y(x) + Derivative(y(x), (x, 3)) - exp(-2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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