7.11.37 problem 38

Internal problem ID [358]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 38
Date solved : Saturday, March 29, 2025 at 04:51:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }+2 y&=\sin \left (3 x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+2*y(x) = sin(3*x); 
ic:=y(0) = 2, D(y)(0) = 0; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (176 \cos \left (x \right )+197 \sin \left (x \right )\right ) {\mathrm e}^{-x}}{85}-\frac {6 \cos \left (3 x \right )}{85}-\frac {7 \sin \left (3 x \right )}{85} \]

✓ Mathematica. Time used: 0.271 (sec). Leaf size: 41

ode=D[y[x],{x,2}]+2*D[y[x],{x,1}]+2*y[x]==Sin[3*x]; 
ic={y[0]==2,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{85} e^{-x} \left (197 \sin (x)-7 e^x \sin (3 x)+176 \cos (x)-6 e^x \cos (3 x)\right ) \]

✓ Sympy. Time used: 0.246 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \left (\frac {197 \sin {\left (x \right )}}{85} + \frac {176 \cos {\left (x \right )}}{85}\right ) e^{- x} - \frac {7 \sin {\left (3 x \right )}}{85} - \frac {6 \cos {\left (3 x \right )}}{85} \]