67.7.9 problem Problem 5(a)

Internal problem ID [14049]
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 : Monday, March 31, 2025 at 08:23:00 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.153 (sec). Leaf size: 30

ode:=[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]; 
ic:=x(0) = 1y(0) = -1; 
dsolve([ode,ic]);
 

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

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

ode={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}; 
ic={x[0]==1,y[0]==-1}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.274 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-7*x(t) - y(t) + 6*exp(t) + Derivative(x(t), t) + 1,0),Eq(4*x(t) - 3*y(t) - 4*exp(t) + Derivative(y(t), t) + 3,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{5 t} + \left (C_{1} + 2 C_{2}\right ) e^{5 t} + e^{t}, \ y{\left (t \right )} = - 4 C_{1} t e^{5 t} - 4 C_{2} e^{5 t} + 1\right ] \]