76.6.11 problem 11

Internal problem ID [17395]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 11
Date solved : Monday, March 31, 2025 at 04:12:45 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+2 \sin \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.253 (sec). Leaf size: 19

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+2*sin(t)]; 
ic:=x(0) = 0y(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= \sin \left (t \right )-\cos \left (t \right ) t \\ y \left (t \right ) &= \sin \left (t \right ) t \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 87

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

\begin{align*} x(t)\to -\cos (t) \int _1^0-2 \sin ^2(K[1])dK[1]+\cos (t) \int _1^t-2 \sin ^2(K[1])dK[1]+\sin ^3(t) \\ y(t)\to \sin (t) \left (-\int _1^t-2 \sin ^2(K[1])dK[1]+\int _1^0-2 \sin ^2(K[1])dK[1]+\sin (t) \cos (t)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.109 (sec). Leaf size: 48

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

\[ \left [ x{\left (t \right )} = C_{1} \sin {\left (t \right )} + C_{2} \cos {\left (t \right )} - t \cos {\left (t \right )} + \sin ^{3}{\left (t \right )} + \sin {\left (t \right )} \cos ^{2}{\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )} + t \sin {\left (t \right )}\right ] \]