83.20.2 problem 5

Internal problem ID [22060]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Examination VII. page 256
Problem number : 5
Date solved : Thursday, October 02, 2025 at 08:23:17 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )+y \left (t \right )-\frac {d}{d t}x \left (t \right )+x \left (t \right )&=t\\ \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+x \left (t \right )-y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 34

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

\begin{align*} x \left (t \right ) &= \sin \left (t \right ) c_2 +\cos \left (t \right ) c_1 -\frac {1}{2}+\frac {t}{2} \\ y \left (t \right ) &= \cos \left (t \right ) c_2 -\sin \left (t \right ) c_1 +\frac {1}{2}+\frac {t}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 25

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

\begin{align*} x(t)&\to \frac {t}{2}+c_1 e^t\\ y(t)&\to \frac {t}{2} \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 85

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

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