81.14.18 problem 18-31

Internal problem ID [21703]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 18. Algebra of differential operators. Page 435
Problem number : 18-31
Date solved : Thursday, October 02, 2025 at 08:00:09 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y+4 y^{\prime }+y^{\prime \prime }+y^{\prime \prime \prime }&=\cos \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 39

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

\[ y = \frac {\left (-20 x +200 c_1 +13\right ) \cos \left (2 x \right )}{200}+\frac {\left (8+5 x +100 c_3 \right ) \sin \left (2 x \right )}{100}+c_2 \,{\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.037 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to c_3 e^{-x}+\frac {1}{50} (5 x+8) \sin (x) \cos (x)+\left (-\frac {x}{10}+\frac {13}{200}+c_1\right ) \cos (2 x)+c_2 \sin (2 x) \end{align*}

✓ Sympy. Time used: 0.242 (sec). Leaf size: 189

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*y(x) - cos(2*x) + 5*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} e^{\frac {x \left (-10 + \frac {25}{\sqrt [3]{12 \sqrt {114} + 179}} + \sqrt [3]{12 \sqrt {114} + 179}\right )}{6}} \sin {\left (\frac {\sqrt {3} x \left (- \sqrt [3]{12 \sqrt {114} + 179} + \frac {25}{\sqrt [3]{12 \sqrt {114} + 179}}\right )}{6} \right )} + C_{2} e^{\frac {x \left (-10 + \frac {25}{\sqrt [3]{12 \sqrt {114} + 179}} + \sqrt [3]{12 \sqrt {114} + 179}\right )}{6}} \cos {\left (\frac {\sqrt {3} x \left (- \sqrt [3]{12 \sqrt {114} + 179} + \frac {25}{\sqrt [3]{12 \sqrt {114} + 179}}\right )}{6} \right )} + C_{3} e^{- \frac {x \left (\frac {25}{\sqrt [3]{12 \sqrt {114} + 179}} + 5 + \sqrt [3]{12 \sqrt {114} + 179}\right )}{3}} - \frac {\sin {\left (2 x \right )}}{40} - \frac {\cos {\left (2 x \right )}}{20} \]