85.67.4 problem 4

Internal problem ID [22895]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 216
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:16:22 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-4 y&=4 x +2+3 \,{\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 61

ode:=diff(diff(diff(y(x),x),x),x)-4*y(x) = 4*x+2+3*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{-\frac {2^{{2}/{3}} x}{2}} \cos \left (\frac {\sqrt {3}\, 2^{{2}/{3}} x}{2}\right )+c_3 \,{\mathrm e}^{-\frac {2^{{2}/{3}} x}{2}} \sin \left (\frac {\sqrt {3}\, 2^{{2}/{3}} x}{2}\right )+c_1 \,{\mathrm e}^{2^{{2}/{3}} x}-\frac {{\mathrm e}^{-2 x}}{4}-x -\frac {1}{2} \]

✓ Mathematica. Time used: 3.384 (sec). Leaf size: 85

ode=D[y[x],{x,3}]-4*y[x]==4*x+2+3*Exp[-2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -x-\frac {e^{-2 x}}{4}+c_3 e^{2^{2/3} x}+c_1 e^{-\frac {x}{\sqrt [3]{2}}} \cos \left (\frac {\sqrt {3} x}{\sqrt [3]{2}}\right )+c_2 e^{-\frac {x}{\sqrt [3]{2}}} \sin \left (\frac {\sqrt {3} x}{\sqrt [3]{2}}\right )-\frac {1}{2} \end{align*}

✓ Sympy. Time used: 0.147 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x - 4*y(x) + Derivative(y(x), (x, 3)) - 2 - 3*exp(-2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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