52.9.5 problem 5

Internal problem ID [8383]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.1. Page 332
Problem number : 5
Date solved : Sunday, March 30, 2025 at 12:53:51 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-y \left (t \right )+z \left (t \right )+t -1\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+y \left (t \right )-z \left (t \right )-3 t^{2}\\ \frac {d}{d t}z \left (t \right )&=x \left (t \right )+y \left (t \right )+z \left (t \right )+t^{2}-t +2 \end{align*}

✓ Maple. Time used: 0.227 (sec). Leaf size: 171

ode:=[diff(x(t),t) = x(t)-y(t)+z(t)+t-1, diff(y(t),t) = 2*x(t)+y(t)-z(t)-3*t^2, diff(z(t),t) = x(t)+y(t)+z(t)+t^2-t+2]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= t^{2}-\frac {1}{6}+c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )+c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right ) \\ y \left (t \right ) &= -\frac {t^{2}}{2}-\frac {7}{4}+\frac {c_1 \,{\mathrm e}^{2 t}}{2}-\frac {c_2 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}+\frac {c_2 \,{\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {c_3 \,{\mathrm e}^{\frac {t}{2}} \sqrt {11}\, \cos \left (\frac {\sqrt {11}\, t}{2}\right )}{2}-\frac {3 t}{2} \\ z \left (t \right ) &= -\frac {t}{2}+\frac {3 c_1 \,{\mathrm e}^{2 t}}{2}-c_2 \,{\mathrm e}^{\frac {t}{2}} \cos \left (\frac {\sqrt {11}\, t}{2}\right )-c_3 \,{\mathrm e}^{\frac {t}{2}} \sin \left (\frac {\sqrt {11}\, t}{2}\right )-\frac {3 t^{2}}{2}-\frac {7}{12} \\ \end{align*}

✓ Mathematica. Time used: 12.552 (sec). Leaf size: 304

ode={D[x[t],t]==x[t]-y[t]+z[t]+t-1,D[y[t],t]==2*x[t]+y[t]-z[t]-3*t^2,D[z[t],t]==x[t]+y[t]+z[t]+t^2-t+2}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t],z[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to t^2+\frac {2}{5} c_1 e^{2 t}+\frac {2}{5} c_3 e^{2 t}+\frac {1}{5} (3 c_1-2 c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )-\frac {(c_1+10 c_2-4 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )}{5 \sqrt {11}}-\frac {1}{6} \\ y(t)\to \frac {1}{220} \left (-11 \left (10 t^2+30 t-4 (c_1+c_3) e^{2 t}+35\right )-44 (c_1-5 c_2+c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )+4 \sqrt {11} (17 c_1+5 c_2-13 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )\right ) \\ z(t)\to -\frac {3 t^2}{2}-\frac {t}{2}+\frac {3}{5} (c_1+c_3) e^{2 t}-\frac {1}{5} (3 c_1-2 c_3) e^{t/2} \cos \left (\frac {\sqrt {11} t}{2}\right )+\frac {(c_1+10 c_2-4 c_3) e^{t/2} \sin \left (\frac {\sqrt {11} t}{2}\right )}{5 \sqrt {11}}-\frac {7}{12} \\ \end{align*}

✓ Sympy. Time used: 1.351 (sec). Leaf size: 491

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-t - x(t) + y(t) - z(t) + Derivative(x(t), t) + 1,0),Eq(3*t**2 - 2*x(t) - y(t) + z(t) + Derivative(y(t), t),0),Eq(-t**2 + t - x(t) - y(t) - z(t) + Derivative(z(t), t) - 2,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {11} t}{2} \right )} - C_{2} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {11} t}{2} \right )} + \frac {2 C_{3} e^{2 t}}{3} + \frac {6 t^{2} \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} + \frac {6 t^{2} \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {t^{2}}{5} + \frac {t \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} + \frac {t \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {t}{5} + \frac {2 \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{15} + \frac {2 \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{15} - \frac {3}{10}, \ y{\left (t \right )} = \frac {C_{3} e^{2 t}}{3} - \frac {2 t^{2} \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {2 t^{2} \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {t^{2}}{10} - \frac {7 t \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {7 t \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {t}{10} - \left (\frac {C_{1}}{2} + \frac {\sqrt {11} C_{2}}{2}\right ) e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {11} t}{2} \right )} - \left (\frac {\sqrt {11} C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {11} t}{2} \right )} - \frac {8 \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {8 \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {3}{20}, \ z{\left (t \right )} = - C_{1} e^{\frac {t}{2}} \sin {\left (\frac {\sqrt {11} t}{2} \right )} + C_{2} e^{\frac {t}{2}} \cos {\left (\frac {\sqrt {11} t}{2} \right )} + C_{3} e^{2 t} - \frac {6 t^{2} \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {6 t^{2} \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {3 t^{2}}{10} - \frac {t \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {t \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{5} - \frac {3 t}{10} - \frac {2 \sin ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{15} - \frac {2 \cos ^{2}{\left (\frac {\sqrt {11} t}{2} \right )}}{15} - \frac {9}{20}\right ] \]