64.17.3 problem 3

Internal problem ID [13552]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.3. Exercises page 299
Problem number : 3
Date solved : Monday, March 31, 2025 at 08:01:08 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+2 y \left (t \right )+5 t\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )+4 y \left (t \right )+17 t \end{align*}

✓ Maple. Time used: 0.133 (sec). Leaf size: 40

ode:=[diff(x(t),t) = 5*x(t)+2*y(t)+5*t, diff(y(t),t) = 3*x(t)+4*y(t)+17*t]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} c_2 +{\mathrm e}^{7 t} c_1 +t +1 \\ y \left (t \right ) &= -\frac {3 \,{\mathrm e}^{2 t} c_2}{2}+{\mathrm e}^{7 t} c_1 -2-5 t \\ \end{align*}

✓ Mathematica. Time used: 0.162 (sec). Leaf size: 234

ode={D[x[t],t]==5*x[t]+2*y[t]+5*t,D[y[t],t]==3*x[t]+4*y[t]+17*t}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{2 t} \left (\left (3 e^{5 t}+2\right ) \int _1^t-\frac {1}{5} e^{-7 K[1]} \left (-49+24 e^{5 K[1]}\right ) K[1]dK[1]+2 \left (e^{5 t}-1\right ) \int _1^t\frac {1}{5} e^{-7 K[2]} \left (49+36 e^{5 K[2]}\right ) K[2]dK[2]+3 c_1 e^{5 t}+2 c_2 e^{5 t}+2 c_1-2 c_2\right ) \\ y(t)\to \frac {1}{5} e^{2 t} \left (3 \left (e^{5 t}-1\right ) \int _1^t-\frac {1}{5} e^{-7 K[1]} \left (-49+24 e^{5 K[1]}\right ) K[1]dK[1]+\left (2 e^{5 t}+3\right ) \int _1^t\frac {1}{5} e^{-7 K[2]} \left (49+36 e^{5 K[2]}\right ) K[2]dK[2]+3 c_1 e^{5 t}+2 c_2 e^{5 t}-3 c_1+3 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.161 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*t - 5*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-17*t - 3*x(t) - 4*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {2 C_{1} e^{2 t}}{3} + C_{2} e^{7 t} + t + 1, \ y{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{7 t} - 5 t - 2\right ] \]