64.16.11 problem 11

Internal problem ID [13543]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 7, Systems of linear differential equations. Section 7.1. Exercises page 277
Problem number : 11
Date solved : Monday, March 31, 2025 at 08:00:53 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.121 (sec). Leaf size: 39

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} c_2 +{\mathrm e}^{-2 t} c_1 -t +1 \\ y \left (t \right ) &= -{\mathrm e}^{4 t} c_2 +{\mathrm e}^{-2 t} c_1 +t \\ \end{align*}

✓ Mathematica. Time used: 0.116 (sec). Leaf size: 208

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

\begin{align*} x(t)\to \frac {1}{2} e^{-2 t} \left (\left (e^{6 t}+1\right ) \int _1^te^{-4 K[1]} \left (4 K[1]+e^{6 K[1]}-3\right )dK[1]-\left (e^{6 t}-1\right ) \int _1^te^{-4 K[2]} \left (-4 K[2]+e^{6 K[2]}+3\right )dK[2]+c_1 e^{6 t}-c_2 e^{6 t}+c_1+c_2\right ) \\ y(t)\to \frac {1}{2} e^{-2 t} \left (-\left (e^{6 t}-1\right ) \int _1^te^{-4 K[1]} \left (4 K[1]+e^{6 K[1]}-3\right )dK[1]+\left (e^{6 t}+1\right ) \int _1^te^{-4 K[2]} \left (-4 K[2]+e^{6 K[2]}+3\right )dK[2]+c_1 \left (-e^{6 t}\right )+c_2 e^{6 t}+c_1+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 36

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

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