23.4.6 problem 8(f)

Internal problem ID [4171]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 8(f)
Date solved : Monday, January 27, 2025 at 08:41:15 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 30

dsolve([diff(y__1(x),x)=y__2(x),diff(y__2(x),x)=y__1(x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 68

DSolve[{D[y1[x],x]==y2[x],D[y2[x],x]==y1[x]},{y1[x],y2[x]},x,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}+1\right )+c_2 \left (e^{2 x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}-1\right )+c_2 \left (e^{2 x}+1\right )\right ) \\ \end{align*}