67.6.5 problem Problem 4(e)

Internal problem ID [14109]
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(e)
Date solved : Tuesday, January 28, 2025 at 06:14:34 AM
CAS classification : system_of_ODEs

\begin{align*} 5 x^{\prime }\left (t \right )-3 y^{\prime }&=x \left (t \right )+y\\ 3 x^{\prime }\left (t \right )-y^{\prime }&=t \end{align*}

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 44

dsolve([5*diff(x(t),t)-3*diff(y(t),t)=x(t)+y(t),3*diff(x(t),t)-diff(y(t),t)=t],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {t^{2}}{8}-{\mathrm e}^{-t} c_{1} +\frac {t}{2}+c_{2} \\ y &= \frac {3 t}{2}-3 \,{\mathrm e}^{-t} c_{1} -2-\frac {t^{2}}{8}-c_{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.084 (sec). Leaf size: 179

DSolve[{5*D[x[t],t]-3*D[y[t],t]==x[t]+y[t],3*D[x[t],t]-D[y[t],t]==t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (\left (3 e^t+1\right ) \int _1^t\frac {1}{4} \left (1+2 e^{K[1]}\right ) K[1]dK[1]-\left (e^t-1\right ) \int _1^t\frac {1}{4} \left (-1+6 e^{K[2]}\right ) K[2]dK[2]+3 c_1 e^t-c_2 e^t+c_1+c_2\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (-3 \left (e^t-1\right ) \int _1^t\frac {1}{4} \left (1+2 e^{K[1]}\right ) K[1]dK[1]+\left (e^t+3\right ) \int _1^t\frac {1}{4} \left (-1+6 e^{K[2]}\right ) K[2]dK[2]-3 c_1 e^t+c_2 e^t+3 c_1+3 c_2\right ) \\ \end{align*}