73.27.15 problem 38.10 (i)

Internal problem ID [15771]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (i)
Date solved : Tuesday, January 28, 2025 at 08:06:55 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+3 y \left (t \right )-6 \,{\mathrm e}^{3 t}\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+6 y \left (t \right )+2 \,{\mathrm e}^{3 t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 4\\ y \left (0\right ) = 0 \end{align*}

✓ Solution by Maple

Time used: 0.065 (sec). Leaf size: 43

dsolve([diff(x(t),t) = 4*x(t)+3*y(t)-6*exp(3*t), diff(y(t),t) = x(t)+6*y(t)+2*exp(3*t), x(0) = 4, y(0) = 0], singsol=all)
 

\begin{align*} x \left (t \right ) &= 3 \,{\mathrm e}^{3 t}+{\mathrm e}^{7 t}-6 t \,{\mathrm e}^{3 t} \\ y \left (t \right ) &= -{\mathrm e}^{3 t}+{\mathrm e}^{7 t}+2 t \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 50

DSolve[{D[x[t],t]==4*x[t]+3*y[t]+6*Exp[3*t],D[y[t],t]==x[t]+6*y[t]+2*Exp[3*t]},{x[0]==4,y[0]==0},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{4} e^{3 t} \left (12 t+7 e^{4 t}+9\right ) \\ y(t)\to \frac {1}{4} e^{3 t} \left (-4 t+7 e^{4 t}-7\right ) \\ \end{align*}