64.17.3 problem 3

Internal problem ID [13623]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.3. Exercises page 299
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 05:54:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+2 y \left (t \right )+5 t\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+4 y \left (t \right )+17 t \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 40

dsolve([diff(x(t),t)=5*x(t)+2*y(t)+5*t,diff(y(t),t)=3*x(t)+4*y(t)+17*t],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{7 t}+c_{1} {\mathrm e}^{2 t}+t +1 \\ y \left (t \right ) &= c_{2} {\mathrm e}^{7 t}-\frac {3 c_{1} {\mathrm e}^{2 t}}{2}-2-5 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.162 (sec). Leaf size: 234

DSolve[{D[x[t],t]==5*x[t]+2*y[t]+5*t,D[y[t],t]==3*x[t]+4*y[t]+17*t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{2 t} \left (\left (3 e^{5 t}+2\right ) \int _1^t-\frac {1}{5} e^{-7 K[1]} \left (-49+24 e^{5 K[1]}\right ) K[1]dK[1]+2 \left (e^{5 t}-1\right ) \int _1^t\frac {1}{5} e^{-7 K[2]} \left (49+36 e^{5 K[2]}\right ) K[2]dK[2]+3 c_1 e^{5 t}+2 c_2 e^{5 t}+2 c_1-2 c_2\right ) \\ y(t)\to \frac {1}{5} e^{2 t} \left (3 \left (e^{5 t}-1\right ) \int _1^t-\frac {1}{5} e^{-7 K[1]} \left (-49+24 e^{5 K[1]}\right ) K[1]dK[1]+\left (2 e^{5 t}+3\right ) \int _1^t\frac {1}{5} e^{-7 K[2]} \left (49+36 e^{5 K[2]}\right ) K[2]dK[2]+3 c_1 e^{5 t}+2 c_2 e^{5 t}-3 c_1+3 c_2\right ) \\ \end{align*}