67.7.12 problem Problem 5(d)

Internal problem ID [14123]
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(d)
Date solved : Tuesday, January 28, 2025 at 06:14:50 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-7 x \left (t \right )+4 y+6 \,{\mathrm e}^{3 t}\\ y^{\prime }&=-5 x \left (t \right )+2 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: 0.072 (sec). Leaf size: 57

dsolve([diff(x(t),t) = -7*x(t)+4*y(t)+6*exp(3*t), diff(y(t),t) = -5*x(t)+2*y(t)+6*exp(2*t), x(0) = 1, y(0) = -1], singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {44 \,{\mathrm e}^{-3 t}}{5}-\frac {46 \,{\mathrm e}^{-2 t}}{5}+\frac {{\mathrm e}^{3 t}}{5}+\frac {6 \,{\mathrm e}^{2 t}}{5} \\ y &= -{\mathrm e}^{3 t}+\frac {44 \,{\mathrm e}^{-3 t}}{5}-\frac {23 \,{\mathrm e}^{-2 t}}{2}+\frac {27 \,{\mathrm e}^{2 t}}{10} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.061 (sec). Leaf size: 66

DSolve[{D[x[t],t]==-7*x[t]+4*y[t]+6*Exp[3*t],D[y[t],t]==-5*x[t]+2*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}{5} e^{-3 t} \left (-46 e^t+6 e^{5 t}+e^{6 t}+44\right ) \\ y(t)\to \frac {1}{10} e^{-3 t} \left (-115 e^t+27 e^{5 t}-10 e^{6 t}+88\right ) \\ \end{align*}