76.10.10 problem 10

Internal problem ID [17532]
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 : 10
Date solved : Tuesday, January 28, 2025 at 08:27:41 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+y \left (t \right )+x \left (t \right )^{2}\\ \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: 0.286 (sec). Leaf size: 77

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

\begin{align*} \left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{\sqrt {\textit {\_a}^{4}-2 \textit {\_a}^{3}+\textit {\_a}^{2}+c_{1}}}d \textit {\_a} +t +c_{2} \right ), x \left (t \right ) = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {1}{\sqrt {\textit {\_a}^{4}-2 \textit {\_a}^{3}+\textit {\_a}^{2}+c_{1}}}d \textit {\_a} +t +c_{2} \right )\right \} \\ \{y \left (t \right ) &= -x \left (t \right )^{2}+\frac {d}{d t}x \left (t \right )+x \left (t \right )\} \\ \end{align*}

✓ Solution by Mathematica

Time used: 3.581 (sec). Leaf size: 6795

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

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