67.7.11 problem Problem 5(c)

Internal problem ID [14122]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 8.3 Systems of Linear Differential Equations (Variation of Parameters). Problems page 514
Problem number : Problem 5(c)
Date solved : Tuesday, January 28, 2025 at 06:14:49 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=7 x \left (t \right )-4 y+10 \,{\mathrm e}^{t}\\ y^{\prime }&=3 x \left (t \right )+14 y+6 \,{\mathrm e}^{2 t} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -1 \end{align*}

✓ Solution by Maple

Time used: 0.076 (sec). Leaf size: 53

dsolve([diff(x(t),t) = 7*x(t)-4*y(t)+10*exp(t), diff(y(t),t) = 3*x(t)+14*y(t)+6*exp(2*t), x(0) = 1, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {67 \,{\mathrm e}^{10 t}}{9}-\frac {14 \,{\mathrm e}^{11 t}}{3}-\frac {{\mathrm e}^{2 t}}{3}-\frac {13 \,{\mathrm e}^{t}}{9} \\ y &= -\frac {67 \,{\mathrm e}^{10 t}}{12}+\frac {14 \,{\mathrm e}^{11 t}}{3}-\frac {5 \,{\mathrm e}^{2 t}}{12}+\frac {{\mathrm e}^{t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 64

DSolve[{D[x[t],t]==7*x[t]-4*y[t]+10*Exp[t],D[y[t],t]==3*x[t]+14*y[t]+6*Exp[2*t]},{x[0]==1,y[0]==-1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to -\frac {1}{9} e^t \left (3 e^t-67 e^{9 t}+42 e^{10 t}+13\right ) \\ y(t)\to \frac {1}{12} e^t \left (-5 e^t-67 e^{9 t}+56 e^{10 t}+4\right ) \\ \end{align*}