problem 1

Internal problem ID [17889]
Book : Diﬀerential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 7. Nonlinear Diﬀerential Equations and Stability. Section 7.1 (Autonomous Systems and Stability). Problems at page 464
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 08:28:11 PM
CAS classiﬁcation : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-2 y \left (t \right )+x \left (t \right ) y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+4 x \left (t \right ) y \left (t \right ) \end{align*}

\begin{align*} \left [\{x \left (t \right ) = 2\}, \left \{y \left (t \right ) &= -\frac {1}{4}+{\mathrm e}^{8 t} c_{1}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {4 \,{\mathrm e}^{\operatorname {RootOf}\left (-\ln \left (\frac {{\mathrm e}^{\textit {\_Z}}}{2}+\frac {9}{2}\right ) {\mathrm e}^{\textit {\_Z}}-32 \,{\mathrm e}^{\textit {\_Z}} \ln \left (\textit {\_a} -2\right )+3 c_{1} {\mathrm e}^{\textit {\_Z}}+\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-16 \textit {\_a} \,{\mathrm e}^{\textit {\_Z}}+9\right )}}{9 \left (\textit {\_a} -2\right )}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}\right \}\right ] \\ \end{align*}

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{4 K[1]+1}dK[1]\&\right ]\left [\int _1^{x(t)}\frac {K[2]}{K[2]-2}dK[2]+c_1\right ] \\ \text {Solve}\left [\int _1^{x(t)}\frac {1}{(K[3]-2) \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {K[1]}{4 K[1]+1}dK[1]\&\right ]\left [c_1+\int _1^{K[3]}\frac {K[2]}{K[2]-2}dK[2]\right ]}dK[3]&=t+c_2,x(t)\right ] \\ \end{align*}

76.30.1 problem 1