61.6.1 problem Problem 4(a)

Internal problem ID [15386]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 6.4 Reduction to a single ODE. Problems page 415
Problem number : Problem 4(a)
Date solved : Thursday, October 02, 2025 at 10:12:27 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 63

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

\begin{align*} x(t)&\to e^{t/2} \left (c_1 \cos \left (\frac {t}{2}\right )+c_2 \sin \left (\frac {t}{2}\right )\right )\\ y(t)&\to e^{t/2} \left (c_2 \cos \left (\frac {t}{2}\right )-c_1 \sin \left (\frac {t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.065 (sec). Leaf size: 51

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

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