60.10.1 problem 1913

Internal problem ID [11912]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1913
Date solved : Tuesday, January 28, 2025 at 06:24:06 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.189 (sec). Leaf size: 54

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

\begin{align*} \left [\{x \left (t \right ) = 0\}, \left \{y \left (t \right ) &= \frac {1}{-t +c_{1}}\right \}\right ] \\ \left [\left \{x \left (t \right ) &= \frac {\tanh \left (\frac {c_{2} +t}{c_{1}}\right )}{c_{1}}\right \}, \left \{y \left (t \right ) = -\frac {x \left (t \right )^{2}+\frac {d}{d t}x \left (t \right )}{x \left (t \right )}\right \}\right ] \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 63

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

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