Internal problem ID [12369]
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).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=\frac {x \left (t \right )}{2}+\frac {y}{2}\\ y^{\prime }&=-\frac {x \left (t \right )}{2}+\frac {y}{2} \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
dsolve([diff(x(t),t)+diff(y(t),t)=y(t),diff(x(t),t)-diff(y(t),t)=x(t)],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 63
DSolve[{x'[t]+y'[t]==y[t],x'[t]-y'[t]==x[t]},{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*}