1.371 problem 372
Internal
problem
ID
[9353]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
372
Date
solved
:
Thursday, October 17, 2024 at 02:19:20 PM
CAS
classification
:
[_quadrature]
Solve
\begin{align*} {y^{\prime }}^{2}-4 y^{3}+a y+b&=0 \end{align*}
Solving for the derivative gives these ODE’s to solve
\begin{align*}
\tag{1} y^{\prime }&=\sqrt {4 y^{3}-a y-b} \\
\tag{2} y^{\prime }&=-\sqrt {4 y^{3}-a y-b} \\
\end{align*}
Now each of the above is solved
separately.
Solving Eq. (1)
Since initial conditions \(\left (x_0,y_0\right ) \) are given, then the result can be written as Since unable to evaluate
the integral, and no initial conditions are given, then the result becomes
\[ \int _{}^{y}\frac {1}{\sqrt {4 \tau ^{3}-a \tau -b}}d \tau = x +c_1 \]
Singular solutions
are found by solving
\begin{align*} \sqrt {4 y^{3}-a y -b}&= 0 \end{align*}
for \(y\). This is because we had to divide by this in the above step. This gives the following
singular solution(s), which also have to satisfy the given ODE.
\begin{align*} y = \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\\ y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2}\\ y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2} \end{align*}
We now need to find the singular solutions, these are found by finding for what values \((\sqrt {4 y^{3}-a y -b})\) is
zero. These give
\begin{align*}
y&=\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y&=-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2} \\
y&=-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2} \\
\end{align*}
Now we go over each such singular solution and check if it verifies the ode
itself and any initial conditions given. If it does not then the singular solution will not be
used.
The solution \(y = \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\) satisfies the ode and initial conditions.
The solution \(y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2}\) satisfies the ode and initial conditions.
The solution \(y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2}\) satisfies the ode and initial conditions.
Solving Eq. (2)
Since initial conditions \(\left (x_0,y_0\right ) \) are given, then the result can be written as Since unable to evaluate
the integral, and no initial conditions are given, then the result becomes
\[ \int _{}^{y}-\frac {1}{\sqrt {4 \tau ^{3}-a \tau -b}}d \tau = x +c_2 \]
Singular solutions
are found by solving
\begin{align*} -\sqrt {4 y^{3}-a y -b}&= 0 \end{align*}
for \(y\). This is because we had to divide by this in the above step. This gives the following
singular solution(s), which also have to satisfy the given ODE.
\begin{align*} y = \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\\ y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2}\\ y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2} \end{align*}
We now need to find the singular solutions, these are found by finding for what values \((-\sqrt {4 y^{3}-a y -b})\) is
zero. These give
\begin{align*}
y&=\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y&=-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2} \\
y&=-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2} \\
\end{align*}
Now we go over each such singular solution and check if it verifies the ode
itself and any initial conditions given. If it does not then the singular solution will not be
used.
The solution \(y = \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}+\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\) satisfies the ode and initial conditions.
The solution \(y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2}\) satisfies the ode and initial conditions.
The solution \(y = -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{12}-\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}}\right )}{2}\) satisfies the ode and initial conditions.
1.371.1 Maple step by step solution
1.371.2 Maple trace
`Methods for first order ODEs:
*** Sublevel 2 ***
Methods for first order ODEs:
-> Solving 1st order ODE of high degree, 1st attempt
trying 1st order WeierstrassP solution for high degree ODE
<- 1st_order WeierstrassP successful`
1.371.3 Maple dsolve solution
Solving time : 0.093 (sec)
Leaf size : 229
dsolve(diff(y(x),x)^2-4*y(x)^3+a*y(x)+b = 0,
y(x),singsol=all)
\begin{align*}
y &= \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}+3 a}{6 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y &= \frac {-i \sqrt {3}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}+3 i \sqrt {3}\, a -\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}-3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {-i \sqrt {3}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}+3 i \sqrt {3}\, a +\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{2}/{3}}+3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{{1}/{3}}} \\
y &= \operatorname {WeierstrassP}\left (c_{1} +x , a , b\right ) \\
\end{align*}
1.371.4 Mathematica DSolve solution
Solving time : 0.49 (sec)
Leaf size : 273
DSolve[{b + a*y[x] - 4*y[x]^3 + D[y[x],x]^2==0,{}},
y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \wp (x-c_1;a,b) \\
y(x)\to \wp (x+c_1;a,b) \\
y(x)\to \frac {\left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}+\sqrt [3]{3} a}{2\ 3^{2/3} \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \\
y(x)\to \frac {i \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}-\sqrt [6]{3} \left (\sqrt {3}+3 i\right ) a}{12 \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \\
y(x)\to \frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (\sqrt {81 b^2-3 a^3}+9 b\right )^{2/3}-\sqrt [6]{3} \left (\sqrt {3}-3 i\right ) a}{12 \sqrt [3]{\sqrt {81 b^2-3 a^3}+9 b}} \\
\end{align*}