14.28.4 problem 8
Internal
problem
ID
[2796]
Book
:
Differential
equations
and
their
applications,
4th
ed.,
M.
Braun
Section
:
Chapter
4.
Qualitative
theory
of
differential
equations.
Section
4.1
(Introduction).
Page
377
Problem
number
:
8
Date
solved
:
Tuesday, January 28, 2025 at 02:38:53 PM
CAS
classification
:
system_of_ODEs
\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )-y^{2}\\ y^{\prime }&=x \left (t \right )^{2}-y\\ z^{\prime }\left (t \right )&={\mathrm e}^{z \left (t \right )}-x \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 0.679 (sec). Leaf size: 568
dsolve([diff(x(t),t)=x(t)-y(t)^2,diff(y(t),t)=x(t)^2-y(t),diff(z(t),t)=exp(z(t))-x(t)],singsol=all)
\begin{align*}
\left \{y &= \operatorname {RootOf}\left (-4 \left (\int _{}^{\textit {\_Z}}\frac {\left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{2}/{3}}}{\left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{4}/{3}}+4 \textit {\_f} \left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{2}/{3}}+16 \textit {\_f}^{2}}d \textit {\_f} \right )+t +c_3 \right ), y = \operatorname {RootOf}\left (-8 \left (\int _{}^{\textit {\_Z}}\frac {\left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{2}/{3}}}{i \left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{4}/{3}} \sqrt {3}-\left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{4}/{3}}-16 i \sqrt {3}\, \textit {\_f}^{2}+8 \textit {\_f} \left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{2}/{3}}-16 \textit {\_f}^{2}}d \textit {\_f} \right )+t +c_3 \right ), y = \operatorname {RootOf}\left (8 \left (\int _{}^{\textit {\_Z}}\frac {\left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{2}/{3}}}{i \left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{4}/{3}} \sqrt {3}+\left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{4}/{3}}-16 i \sqrt {3}\, \textit {\_f}^{2}-8 \textit {\_f} \left (-4 \textit {\_f}^{3}+6 c_2 +2 \sqrt {4 \textit {\_f}^{6}-12 c_2 \,\textit {\_f}^{3}-16 \textit {\_f}^{3}+9 c_2^{2}}\right )^{{2}/{3}}+16 \textit {\_f}^{2}}d \textit {\_f} \right )+t +c_3 \right )\right \} \\
\left \{x \left (t \right ) &= \sqrt {y^{\prime }+y}, x \left (t \right ) = -\sqrt {y^{\prime }+y}\right \} \\
\{z \left (t \right ) &= -\int x \left (t \right )d t -\ln \left (-c_1 -\int {\mathrm e}^{\int -x \left (t \right )d t}d t \right )\} \\
\end{align*}
✓ Solution by Mathematica
Time used: 23.461 (sec). Leaf size: 20958
DSolve[{D[x[t],t]==x[t]-y[t]^2,D[y[t],t]==x[t]^2-y[t],D[z[t],t]==Exp[z[t]]-x[t]},{x[t],y[t],z[t]},t,IncludeSingularSolutions -> True]
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