83.50.3 problem Ex 6 page 150

Internal problem ID [19570]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Book Solved Excercises. Chapter IX. Simultaneous equations
Problem number : Ex 6 page 150
Date solved : Monday, March 31, 2025 at 07:33:48 PM
CAS classification : system_of_ODEs

\begin{align*} 4 \frac {d}{d t}x \left (t \right )+9 \frac {d}{d t}y \left (t \right )+11 x \left (t \right )+31 y \left (t \right )&={\mathrm e}^{t}\\ 3 \frac {d}{d t}x \left (t \right )+7 \frac {d}{d t}y \left (t \right )+8 x \left (t \right )+24 y \left (t \right )&={\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.204 (sec). Leaf size: 64

ode:=[4*diff(x(t),t)+9*diff(y(t),t)+11*x(t)+31*y(t) = exp(t), 3*diff(x(t),t)+7*diff(y(t),t)+8*x(t)+24*y(t) = exp(2*t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-4 t} c_2 +{\mathrm e}^{-4 t} t c_1 +\frac {31 \,{\mathrm e}^{t}}{25}-\frac {49 \,{\mathrm e}^{2 t}}{36} \\ y \left (t \right ) &= \frac {19 \,{\mathrm e}^{2 t}}{36}-{\mathrm e}^{-4 t} c_2 -{\mathrm e}^{-4 t} t c_1 -{\mathrm e}^{-4 t} c_1 -\frac {11 \,{\mathrm e}^{t}}{25} \\ \end{align*}

✓ Mathematica. Time used: 0.219 (sec). Leaf size: 76

ode={4*D[x[t],t]+9*D[y[t],t]+11*x[t]+31*y[t]==Exp[t],3*D[x[t],t]+7*D[y[t],t]+8*x[t]+24*y[t]==Exp[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {31 e^t}{25}-\frac {49 e^{2 t}}{36}-e^{-4 t} (c_1 (t-1)+c_2 t) \\ y(t)\to -\frac {11 e^t}{25}+\frac {19 e^{2 t}}{36}+e^{-4 t} ((c_1+c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.256 (sec). Leaf size: 66

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(11*x(t) + 31*y(t) - exp(t) + 4*Derivative(x(t), t) + 9*Derivative(y(t), t),0),Eq(8*x(t) + 24*y(t) - exp(2*t) + 3*Derivative(x(t), t) + 7*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - C_{2} t e^{- 4 t} - \left (C_{1} - C_{2}\right ) e^{- 4 t} - \frac {49 e^{2 t}}{36} + \frac {31 e^{t}}{25}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} t e^{- 4 t} + \frac {19 e^{2 t}}{36} - \frac {11 e^{t}}{25}\right ] \]