85.86.7 problem 3

Internal problem ID [23024]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 491
Problem number : 3
Date solved : Sunday, October 12, 2025 at 05:55:07 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (\pi \right )&=2 \\ y \left (0\right )&=0 \\ D\left (y \right )\left (0\right )&={\frac {1}{2}} \\ \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 17

ode:=[diff(x(t),t)+diff(y(t),t) = cos(t), x(t)+diff(diff(y(t),t),t) = 2]; 
ic:=[x(Pi) = 2, y(0) = 0, D(y)(0) = 1/2]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.164 (sec). Leaf size: 22

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

\begin{align*} x(t)&\to \frac {1}{2} (\sin (t)+4)\\ y(t)&\to \frac {\sin (t)}{2} \end{align*}

✓ Sympy. Time used: 0.518 (sec). Leaf size: 15

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

\[ \left [ x{\left (t \right )} = \frac {\sin {\left (t \right )}}{2} + 2, \ y{\left (t \right )} = \frac {\sin {\left (t \right )}}{2}\right ] \]