23.4.5 problem 8(e)

Internal problem ID [4170]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 8(e)
Date solved : Monday, January 27, 2025 at 08:41:14 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.024 (sec). Leaf size: 69

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

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 139

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

\begin{align*} \text {y1}(x)\to \frac {1}{4} e^{-\sqrt {2} x} \left (c_1 \left (\left (2+\sqrt {2}\right ) e^{2 \sqrt {2} x}+2-\sqrt {2}\right )+\sqrt {2} c_2 \left (e^{2 \sqrt {2} x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{4} e^{-\sqrt {2} x} \left (\sqrt {2} c_1 \left (e^{2 \sqrt {2} x}-1\right )-c_2 \left (\left (\sqrt {2}-2\right ) e^{2 \sqrt {2} x}-2-\sqrt {2}\right )\right ) \\ \end{align*}