76.10.12 problem 12

Internal problem ID [17534]
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 : 12
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 )&=3 x \left (t \right )-x \left (t \right )^{2}\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right ) y \left (t \right )-3 y \left (t \right )+2 \end{align*}

✓ Solution by Maple

Time used: 0.156 (sec). Leaf size: 49

dsolve([diff(x(t),t)=3*x(t)-x(t)^2,diff(y(t),t)=2*x(t)*y(t)-3*y(t)+2],singsol=all)
 

\begin{align*} \left \{x \left (t \right ) &= \frac {3}{1+3 \,{\mathrm e}^{-3 t} c_{2}}\right \} \\ \{y \left (t \right ) &= \left (\int 2 \,{\mathrm e}^{-\int \left (-3+2 x \left (t \right )\right )d t}d t +c_{1} \right ) {\mathrm e}^{\int \left (-3+2 x \left (t \right )\right )d t}\} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.080 (sec). Leaf size: 133

DSolve[{D[x[t],t]==3*x[t]-x[t]^2,D[y[t],t]==2*x[t]*y[t]-3*y[t]+2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-3) K[1]}dK[1]\&\right ][-t+c_1] \\ y(t)\to \exp \left (\int _1^t\left (2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-3) K[1]}dK[1]\&\right ][c_1-K[2]]-3\right )dK[2]\right ) \left (\int _1^t2 \exp \left (-\int _1^{K[3]}\left (2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-3) K[1]}dK[1]\&\right ][c_1-K[2]]-3\right )dK[2]\right )dK[3]+c_2\right ) \\ \end{align*}