64.11.5 problem 5

Internal problem ID [13376]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 09:50:46 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

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

\[ y = c_{2} {\mathrm e}^{-x} \sin \left (\sqrt {3}\, x \right )+{\mathrm e}^{-x} \cos \left (\sqrt {3}\, x \right ) c_{1} +\frac {\sin \left (4 x \right )}{26}-\frac {3 \cos \left (4 x \right )}{52} \]

✓ Mathematica. Time used: 0.464 (sec). Leaf size: 115

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

\[ y(x)\to e^{-x} \left (\cos \left (\sqrt {3} x\right ) \int _1^x-\frac {e^{K[2]} \cos (4 K[2]) \sin \left (\sqrt {3} K[2]\right )}{\sqrt {3}}dK[2]+\sin \left (\sqrt {3} x\right ) \int _1^x\frac {e^{K[1]} \cos (4 K[1]) \cos \left (\sqrt {3} K[1]\right )}{\sqrt {3}}dK[1]+c_2 \cos \left (\sqrt {3} x\right )+c_1 \sin \left (\sqrt {3} x\right )\right ) \]

✓ Sympy. Time used: 0.248 (sec). Leaf size: 41

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

\[ y{\left (x \right )} = \left (C_{1} \sin {\left (\sqrt {3} x \right )} + C_{2} \cos {\left (\sqrt {3} x \right )}\right ) e^{- x} + \frac {\sin {\left (4 x \right )}}{26} - \frac {3 \cos {\left (4 x \right )}}{52} \]