67.6.1 problem Problem 4(a)

Internal problem ID [14105]
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 : Tuesday, January 28, 2025 at 06:14:31 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*}

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 44

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 &= {\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: 0.003 (sec). Leaf size: 63

DSolve[{D[x[t],t]+D[y[t],t]==y[t],D[x[t],t]-D[y[t],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*}