75.31.1 problem 815

Internal problem ID [17189]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 23.2 The method of undetermined coefficients. Exercises page 239
Problem number : 815
Date solved : Thursday, March 13, 2025 at 09:18:30 AM
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 )&=1-x \left (t \right ) \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 27

ode:=[diff(x(t),t) = y(t), diff(y(t),t) = -x(t)+1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} +1 \\ y \left (t \right ) &= c_{2} \cos \left (t \right )-c_{1} \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 88

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

\begin{align*} x(t)\to \cos (t) \int _1^t-\sin (K[1])dK[1]+\sin (t) \int _1^t\cos (K[2])dK[2]+c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to -\sin (t) \int _1^t-\sin (K[1])dK[1]+\cos (t) \int _1^t\cos (K[2])dK[2]+c_2 \cos (t)-c_1 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 34

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) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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