84.28.8 problem 16.13

Internal problem ID [22285]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 16. Initial-value problems. Supplementary problems
Problem number : 16.13
Date solved : Thursday, October 02, 2025 at 08:37:05 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 33

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)+2*y(x) = sin(2*x)+cos(2*x); 
ic:=[y(0) = 0, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (3 \cos \left (x \right )+11 \sin \left (x \right )\right ) {\mathrm e}^{-x}}{10}-\frac {3 \cos \left (x \right )^{2}}{5}+\frac {\sin \left (x \right ) \cos \left (x \right )}{5}+\frac {3}{10} \]

✓ Mathematica. Time used: 0.121 (sec). Leaf size: 40

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

\begin{align*} y(x)&\to \frac {1}{10} e^{-x} \left (11 \sin (x)-3 e^x \cos (2 x)+\left (2 e^x \sin (x)+3\right ) \cos (x)\right ) \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) - sin(2*x) - cos(2*x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (\frac {11 \sin {\left (x \right )}}{10} + \frac {3 \cos {\left (x \right )}}{10}\right ) e^{- x} - \frac {\sqrt {2} \sin {\left (2 x + \frac {\pi }{4} \right )}}{10} - \frac {\sqrt {2} \cos {\left (2 x + \frac {\pi }{4} \right )}}{5} \]