76.10.13 problem 13

Internal problem ID [17535]
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 : 13
Date solved : Tuesday, January 28, 2025 at 08:27:42 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 2.080 (sec). Leaf size: 70

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

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

✓ Solution by Mathematica

Time used: 0.153 (sec). Leaf size: 94

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

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]-1}{K[1]}dK[1]\&\right ][-2 x(t)-\log (x(t))+c_1] \\ \text {Solve}\left [\int _1^{x(t)}\frac {1}{K[2] \left (\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]-1}{K[1]}dK[1]\&\right ][c_1-2 K[2]-\log (K[2])]-1\right )}dK[2]&=-t+c_2,x(t)\right ] \\ \end{align*}