76.30.2 problem 2
Internal
problem
ID
[17890]
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
:
2
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 )&=1+5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=1-6 x \left (t \right )^{2} \end{align*}
✓ Solution by Maple
Time used: 0.076 (sec). Leaf size: 25
dsolve([diff(x(t),t)=1+5*y(t),diff(y(t),t)=1-6*x(t)^2],singsol=all)
\begin{align*}
\left \{x \left (t \right ) &= -\frac {\operatorname {WeierstrassP}\left (t +c_{1} , 50, c_{2}\right )}{5}\right \} \\
\left \{y \left (t \right ) &= \frac {\frac {d}{d t}x \left (t \right )}{5}-\frac {1}{5}\right \} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.574 (sec). Leaf size: 923
DSolve[{D[x[t],t]==1+5*y[t],D[y[t],t]==1-6*x[t]^2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
y(t)\to \frac {1}{5} \left (-1-\sqrt {-20 x(t)^3+10 x(t)+1+10 c_1}\right ) \\
\text {Solve}\left [\frac {2 \sqrt {\frac {x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,1\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,1\right ]}} \sqrt {\frac {x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,2\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,2\right ]}} \left (x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]-x(t)}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,2\right ]}}\right ),\frac {\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,2\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,1\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]}\right )}{\sqrt {-20 x(t)^3+10 x(t)+1+10 c_1} \sqrt {\frac {x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,2\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]}}}&=-t+c_2,x(t)\right ] \\
y(t)\to \frac {1}{5} \left (-1+\sqrt {-20 x(t)^3+10 x(t)+1+10 c_1}\right ) \\
\text {Solve}\left [\frac {2 \sqrt {\frac {x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,1\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,1\right ]}} \sqrt {\frac {x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,2\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,2\right ]}} \left (x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]-x(t)}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,2\right ]}}\right ),\frac {\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,2\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,1\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-10 c_1-1\&,3\right ]}\right )}{\sqrt {-20 x(t)^3+10 x(t)+1+10 c_1} \sqrt {\frac {x(t)-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]}{\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,2\right ]-\text {Root}\left [20 \text {$\#$1}^3-10 \text {$\#$1}-1-10 c_1\&,3\right ]}}}&=t+c_2,x(t)\right ] \\
\end{align*}