28.2.50 problem 50

Internal problem ID [4493]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 50
Date solved : Tuesday, March 04, 2025 at 06:49:02 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime }-4 y&=50 \sin \left (x \right )+50 \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 38

ode:=diff(diff(diff(y(x),x),x),x)-2*diff(y(x),x)-4*y(x) = 50*sin(x)+50*exp(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \left (c_{1} +5 x -3\right ) {\mathrm e}^{2 x}+\left (c_{2} \cos \left (x \right )+c_3 \sin \left (x \right )\right ) {\mathrm e}^{-x}+6 \cos \left (x \right )-8 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 47

ode=D[y[x],{x,3}]-2*D[y[x],x]-4*y[x]==50*(Sin[x]+Exp[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x} \left (e^{3 x} (5 x-3+c_3)+\left (6 e^x+c_2\right ) \cos (x)+\left (-8 e^x+c_1\right ) \sin (x)\right ) \]

✓ Sympy. Time used: 0.289 (sec). Leaf size: 36

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) - 50*exp(2*x) - 50*sin(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + 5 x\right ) e^{2 x} + \left (C_{2} \sin {\left (x \right )} + C_{3} \cos {\left (x \right )}\right ) e^{- x} - 8 \sin {\left (x \right )} + 6 \cos {\left (x \right )} \]