76.10.19 problem 19

Internal problem ID [17541]
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 : 19
Date solved : Tuesday, January 28, 2025 at 08:27:43 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.339 (sec). Leaf size: 104

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

\begin{align*} \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1}{1+{\mathrm e}^{-t} c_{1}}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}-\frac {3}{\sqrt {-12 \textit {\_a}^{4}+9 c_{1} \textit {\_a} +9 \textit {\_a}^{2}}}d \textit {\_a} +t +c_{2} \right ), x \left (t \right ) = \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {3}{\sqrt {-12 \textit {\_a}^{4}+9 c_{1} \textit {\_a} +9 \textit {\_a}^{2}}}d \textit {\_a} +t +c_{2} \right )\right \}, \left \{y \left (t \right ) = \frac {\frac {d}{d t}x \left (t \right )+x \left (t \right )}{2 x \left (t \right )}\right \}\right ] \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.215 (sec). Leaf size: 995

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

\begin{align*} y(t)\to \frac {1}{2}-\frac {\sqrt {-\frac {2}{3} x(t)^4+\frac {x(t)^2}{2}+2 c_1 x(t)}}{\sqrt {2} x(t)} \\ \text {Solve}\left [-\frac {2 x(t)^2 \sqrt {\frac {\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,1\right ]}{\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,1\right ]}} \sqrt {\frac {\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,2\right ]}{\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,2\right ]}} \left (\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]-\frac {1}{x(t)}}{\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]-\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,2\right ]}}\right ),\frac {\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,2\right ]-\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]}{\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,1\right ]-\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]}\right )}{\sqrt {x(t) \left (-4 x(t)^3+3 x(t)+12 c_1\right )} \sqrt {\frac {\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]}{\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,2\right ]-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]}}}&=-\frac {t}{\sqrt {3}}+c_2,x(t)\right ] \\ y(t)\to \frac {1}{2}+\frac {\sqrt {-\frac {2}{3} x(t)^4+\frac {x(t)^2}{2}+2 c_1 x(t)}}{\sqrt {2} x(t)} \\ \text {Solve}\left [-\frac {2 x(t)^2 \sqrt {\frac {\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,1\right ]}{\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,1\right ]}} \sqrt {\frac {\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,2\right ]}{\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,2\right ]}} \left (\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]-\frac {1}{x(t)}}{\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]-\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,2\right ]}}\right ),\frac {\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,2\right ]-\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]}{\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,1\right ]-\text {Root}\left [12 c_1 \text {$\#$1}^3+3 \text {$\#$1}^2-4\&,3\right ]}\right )}{\sqrt {x(t) \left (-4 x(t)^3+3 x(t)+12 c_1\right )} \sqrt {\frac {\frac {1}{x(t)}-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]}{\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,2\right ]-\text {Root}\left [12 \text {$\#$1}^3 c_1+3 \text {$\#$1}^2-4\&,3\right ]}}}&=\frac {t}{\sqrt {3}}+c_2,x(t)\right ] \\ \end{align*}