14.27.3 problem 3
Internal
problem
ID
[2791]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.1
(Introduction).
Page
3770
Problem
number
:
3
Date
solved
:
Tuesday, January 28, 2025 at 02:38:52 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=a x \left (t \right )-b x \left (t \right ) y\\ y^{\prime }&=-c y+d x \left (t \right ) y\\ z^{\prime }\left (t \right )&=z \left (t \right )+x \left (t \right )^{2}+y^{2} \end{align*}
✓ Solution by Maple
Time used: 0.487 (sec). Leaf size: 171
dsolve([diff(x(t),t)=a*x(t)-b*x(t)*y(t),diff(y(t),t)=-c*y(t)+d*x(t)*y(t),diff(z(t),t)=z(t)+x(t)^2+y(t)^2],singsol=all)
\begin{align*}
\\
\left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{a \textit {\_a} \left (\operatorname {LambertW}\left (\frac {{\mathrm e}^{-1} \textit {\_a}^{-\frac {c}{a}} {\mathrm e}^{\frac {\textit {\_a} d}{a}} {\mathrm e}^{\frac {c_2}{a}}}{a}\right )+1\right )}d \textit {\_a} +t +c_3 \right )\right \}, \left \{y = \frac {a x \left (t \right )-x^{\prime }\left (t \right )}{x \left (t \right ) b}\right \}, \left \{z \left (t \right ) = \left (\int \frac {{\mathrm e}^{-t} \left (x \left (t \right )^{4} b^{2}+a^{2} x \left (t \right )^{2}-2 x^{\prime }\left (t \right ) a x \left (t \right )+{x^{\prime }\left (t \right )}^{2}\right )}{x \left (t \right )^{2} b^{2}}d t +c_1 \right ) {\mathrm e}^{t}\right \}\right ] \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.352 (sec). Leaf size: 459
DSolve[{D[x[t],t]==a*x[t]-b*x[t]*y[t],D[y[t],t]==-c*y[t]+d*x[t]*y[t],D[z[t],t]==z[t]+x[t]^2+y[t]^2},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
y(t)\to -\frac {a W\left (-\frac {b \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][a t+c_2]{}^{-\frac {c}{a}} \exp \left (\frac {d \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][a t+c_2]-c_1}{a}\right )}{a}\right )}{b} \\
x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][a t+c_2] \\
z(t)\to e^t \left (\int _1^t\frac {e^{-K[2]} \left (a^2 W\left (-\frac {b \exp \left (\frac {d \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][c_2+a K[2]]-c_1}{a}\right ) \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][c_2+a K[2]]{}^{-\frac {c}{a}}}{a}\right ){}^2+b^2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (-\frac {b e^{\frac {d K[1]}{a}-\frac {c_1}{a}} K[1]^{-\frac {c}{a}}}{a}\right )+1\right )}dK[1]\&\right ][c_2+a K[2]]{}^2\right )}{b^2}dK[2]+c_3\right ) \\
\end{align*}