64.16.12 problem 12

Internal problem ID [13544]
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 : 12
Date solved : Monday, March 31, 2025 at 08:00:55 AM
CAS classification : system_of_ODEs

\begin{align*} \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 )&=t^{2}\\ \frac {d}{d t}x \left (t \right )+2 \frac {d}{d t}y \left (t \right )-2 x \left (t \right )+4 y \left (t \right )&=2 t +1 \end{align*}

✓ Maple. Time used: 0.154 (sec). Leaf size: 115

ode:=[diff(x(t),t)+diff(y(t),t)-x(t)+5*y(t) = t^2, diff(x(t),t)+2*diff(y(t),t)-2*x(t)+4*y(t) = 2*t+1]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_2 +{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_1 +\frac {2 t^{2}}{3}-\frac {7 t}{9}-\frac {41}{27} \\ y \left (t \right ) &= \frac {t^{2}}{3}-\frac {{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_2}{12}-\frac {{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_2}{12}-\frac {{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {23}\, t}{2}\right ) c_1}{12}+\frac {{\mathrm e}^{\frac {t}{2}} \sqrt {23}\, \sin \left (\frac {\sqrt {23}\, t}{2}\right ) c_1}{12}-\frac {5 t}{9}-\frac {1}{27} \\ \end{align*}

✓ Mathematica. Time used: 6.778 (sec). Leaf size: 540

ode={D[x[t],t]+D[y[t],t]-x[t]+5*y[t]==t^2,D[x[t],t]+2*D[y[t],t]-2*x[t]+4*y[t]==2*t+1}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -\frac {1}{23} e^{t/2} \left (\left (\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )-23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{23} e^{-\frac {K[1]}{2}} \left (23 \cos \left (\frac {1}{2} \sqrt {23} K[1]\right ) \left (2 K[1]^2-2 K[1]-1\right )+\sqrt {23} \left (-10 K[1]^2+22 K[1]+11\right ) \sin \left (\frac {1}{2} \sqrt {23} K[1]\right )\right )dK[1]+12 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right ) \int _1^t\frac {1}{23} e^{-\frac {K[2]}{2}} \left (\sqrt {23} \left (-3 K[2]^2+2 K[2]+1\right ) \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )-23 \cos \left (\frac {1}{2} \sqrt {23} K[2]\right ) \left (K[2]^2-2 K[2]-1\right )\right )dK[2]-23 c_1 \cos \left (\frac {\sqrt {23} t}{2}\right )+\sqrt {23} c_1 \sin \left (\frac {\sqrt {23} t}{2}\right )+12 \sqrt {23} c_2 \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \\ y(t)\to \frac {1}{23} e^{t/2} \left (2 \sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right ) \int _1^t\frac {1}{23} e^{-\frac {K[1]}{2}} \left (23 \cos \left (\frac {1}{2} \sqrt {23} K[1]\right ) \left (2 K[1]^2-2 K[1]-1\right )+\sqrt {23} \left (-10 K[1]^2+22 K[1]+11\right ) \sin \left (\frac {1}{2} \sqrt {23} K[1]\right )\right )dK[1]+\left (\sqrt {23} \sin \left (\frac {\sqrt {23} t}{2}\right )+23 \cos \left (\frac {\sqrt {23} t}{2}\right )\right ) \int _1^t\frac {1}{23} e^{-\frac {K[2]}{2}} \left (\sqrt {23} \left (-3 K[2]^2+2 K[2]+1\right ) \sin \left (\frac {1}{2} \sqrt {23} K[2]\right )-23 \cos \left (\frac {1}{2} \sqrt {23} K[2]\right ) \left (K[2]^2-2 K[2]-1\right )\right )dK[2]+23 c_2 \cos \left (\frac {\sqrt {23} t}{2}\right )+2 \sqrt {23} c_1 \sin \left (\frac {\sqrt {23} t}{2}\right )+\sqrt {23} c_2 \sin \left (\frac {\sqrt {23} t}{2}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 1.032 (sec). Leaf size: 289

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-t**2 - x(t) + 5*y(t) + Derivative(x(t), t) + Derivative(y(t), t),0),Eq(-2*t - 2*x(t) + 4*y(t) + Derivative(x(t), t) + 2*Derivative(y(t), t) - 1,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {2 t^{2} \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{3} + \frac {2 t^{2} \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{3} - \frac {7 t \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{9} - \frac {7 t \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{9} + \left (\frac {C_{1}}{2} - \frac {\sqrt {23} C_{2}}{2}\right ) e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {23} t}{2} \right )} - \left (\frac {\sqrt {23} C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {23} t}{2} \right )} - \frac {41 \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{27} - \frac {41 \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{27}, \ y{\left (t \right )} = - C_{1} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {23} t}{2} \right )} + C_{2} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {23} t}{2} \right )} + \frac {t^{2} \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{3} + \frac {t^{2} \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{3} - \frac {5 t \sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{9} - \frac {5 t \cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{9} - \frac {\sin ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{27} - \frac {\cos ^{2}{\left (\frac {\sqrt {23} t}{2} \right )}}{27}\right ] \]