72.14.2 problem 4

Internal problem ID [14865]
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 : 4
Date solved : Tuesday, January 28, 2025 at 07:17:53 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 76

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

\begin{align*} x(t)\to c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to c_2 \cos (t)-c_1 \sin (t) \\ z(t)\to c_3 e^{2 t} \\ x(t)\to c_1 \cos (t)+c_2 \sin (t) \\ y(t)\to c_2 \cos (t)-c_1 \sin (t) \\ z(t)\to 0 \\ \end{align*}