23.4.9 problem 10

Internal problem ID [4174]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 10
Date solved : Monday, January 27, 2025 at 08:41:18 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=-2 y_{2} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+2 y_{2} \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = -1\\ y_{2} \left (0\right ) = 1 \end{align*}

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 24

dsolve([diff(y__1(x),x) = -2*y__2(x), diff(y__2(x),x) = y__1(x)+2*y__2(x), y__1(0) = -1, y__2(0) = 1], singsol=all)
 

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{x} \left (-\sin \left (x \right )-\cos \left (x \right )\right ) \\ y_{2} \left (x \right ) &= \cos \left (x \right ) {\mathrm e}^{x} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 24

DSolve[{D[y1[x],x]==-2*y2[x],D[y2[x],x]==y1[x]+2*y2[x]},{y1[0]==-1,y2[0]==1},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to -e^x (\sin (x)+\cos (x)) \\ \text {y2}(x)\to e^x \cos (x) \\ \end{align*}