72.15.7 problem 19 (iv)

Internal problem ID [14888]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Review Exercises for chapter 3. page 376
Problem number : 19 (iv)
Date solved : Tuesday, January 28, 2025 at 07:18:10 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 39

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

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