76.10.8 problem 8

Internal problem ID [17530]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.6 (A brief introduction to nonlinear systems). Problems at page 195
Problem number : 8
Date solved : Tuesday, January 28, 2025 at 10:39:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )-x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 67

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

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 137

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

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