67.7.9 problem Problem 5(a)

Internal problem ID [14120]
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(a)
Date solved : Tuesday, January 28, 2025 at 06:14:47 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=7 x \left (t \right )+y-1-6 \,{\mathrm e}^{t}\\ y^{\prime }&=-4 x \left (t \right )+3 y+4 \,{\mathrm e}^{t}-3 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 30

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

\begin{align*} x \left (t \right ) &= -2 t \,{\mathrm e}^{5 t}+{\mathrm e}^{t} \\ y &= 1-{\mathrm e}^{5 t} \left (-4 t +2\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.229 (sec). Leaf size: 368

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

\begin{align*} x(t)\to e^{5 t} \left (t \left (-\int _1^0e^{-5 K[2]} \left (-10 K[2]+4 e^{K[2]} (K[2]+1)-3\right )dK[2]\right )+t \int _1^te^{-5 K[2]} \left (-10 K[2]+4 e^{K[2]} (K[2]+1)-3\right )dK[2]-(2 t+1) \int _1^0e^{-5 K[1]} \left (5 K[1]-e^{K[1]} (2 K[1]+1)-1\right )dK[1]+(2 t+1) \int _1^te^{-5 K[1]} \left (5 K[1]-e^{K[1]} (2 K[1]+1)-1\right )dK[1]+t+1\right ) \\ y(t)\to e^{5 t} \left (4 t \int _1^0e^{-5 K[1]} \left (5 K[1]-e^{K[1]} (2 K[1]+1)-1\right )dK[1]-4 t \int _1^te^{-5 K[1]} \left (5 K[1]-e^{K[1]} (2 K[1]+1)-1\right )dK[1]+2 t \int _1^0e^{-5 K[2]} \left (-10 K[2]+4 e^{K[2]} (K[2]+1)-3\right )dK[2]-2 t \int _1^te^{-5 K[2]} \left (-10 K[2]+4 e^{K[2]} (K[2]+1)-3\right )dK[2]+\int _1^te^{-5 K[2]} \left (-10 K[2]+4 e^{K[2]} (K[2]+1)-3\right )dK[2]-\int _1^0e^{-5 K[2]} \left (-10 K[2]+4 e^{K[2]} (K[2]+1)-3\right )dK[2]-2 t-1\right ) \\ \end{align*}