67.6.4 problem Problem 4(d)

Internal problem ID [14108]
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(d)
Date solved : Tuesday, January 28, 2025 at 06:14:33 AM
CAS classification : system_of_ODEs

\begin{align*} 2 x^{\prime }\left (t \right )-y^{\prime }&=t\\ 3 x^{\prime }\left (t \right )+2 y^{\prime }&=y \end{align*}

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 36

dsolve([2*diff(x(t),t)-diff(y(t),t)=t,3*diff(x(t),t)+2*diff(y(t),t)=y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{4}+\frac {7 \,{\mathrm e}^{\frac {2 t}{7}} c_{1}}{2}+\frac {3 t}{4}+c_{2} \\ y &= \frac {3 t}{2}+7 \,{\mathrm e}^{\frac {2 t}{7}} c_{1} +\frac {21}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.143 (sec). Leaf size: 117

DSolve[{2*D[x[t],t]-D[y[t],t]==t,3*D[x[t],t]+2*D[y[t],t]==y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \int _1^t\frac {1}{14} \left (7-3 e^{-\frac {2 K[1]}{7}}\right ) K[1]dK[1]+\frac {1}{2} \left (e^{2 t/7}-1\right ) \int _1^t-\frac {3}{7} e^{-\frac {2 K[2]}{7}} K[2]dK[2]+\frac {1}{2} c_2 \left (e^{2 t/7}-1\right )+c_1 \\ y(t)\to e^{2 t/7} \left (\int _1^t-\frac {3}{7} e^{-\frac {2 K[2]}{7}} K[2]dK[2]+c_2\right ) \\ \end{align*}