72.14.8 problem 12

Internal problem ID [14871]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.8 page 371
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 07:17:57 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 35

dsolve([diff(x(t),t)=-1*x(t)+2*y(t)+0*z(t),diff(y(t),t)=2*x(t)-4*y(t)+0*z(t),diff(z(t),t)=0*x(t)+0*y(t)-1*z(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 158

DSolve[{D[x[t],t]==-1*x[t]+2*y[t]+0*z[t],D[y[t],t]==2*x[t]-4*y[t]+0*z[t],D[z[t],t]==0*x[t]+0*y[t]-1*z[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{5} e^{-5 t} \left (c_1 \left (4 e^{5 t}+1\right )+2 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-5 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \\ z(t)\to c_3 e^{-t} \\ x(t)\to \frac {1}{5} e^{-5 t} \left (c_1 \left (4 e^{5 t}+1\right )+2 c_2 \left (e^{5 t}-1\right )\right ) \\ y(t)\to \frac {1}{5} e^{-5 t} \left (2 c_1 \left (e^{5 t}-1\right )+c_2 \left (e^{5 t}+4\right )\right ) \\ z(t)\to 0 \\ \end{align*}