84.41.3 problem 31.4

Internal problem ID [22385]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 31. Solutions of linear systems with constant coefficients. Solved problems. Page 194
Problem number : 31.4
Date solved : Thursday, October 02, 2025 at 08:38:07 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (\pi \right )&=1 \\ y \left (\pi \right )&=2 \\ \end{align*}

✓ Maple. Time used: 0.047 (sec). Leaf size: 26

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+3]; 
ic:=[x(Pi) = 1, y(Pi) = 2]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= -2 \sin \left (t \right )+2 \cos \left (t \right )+3 \\ y \left (t \right ) &= -2 \cos \left (t \right )-2 \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 25

ode={D[x[t],t]==y[t],D[y[t],t]==-x[t]+3}; 
ic={x[Pi]==1,y[Pi]==2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -2 \sin (t)+2 \cos (t)+3\\ y(t)&\to -2 (\sin (t)+\cos (t)) \end{align*}

✓ Sympy. Time used: 0.079 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-y(t) + Derivative(x(t), t),0),Eq(x(t) + Derivative(y(t), t) - 3,0)] 
ics = {x(pi): 1, y(pi): 2} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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