60.7.185 problem 1805 (book 6.214)
Internal
problem
ID
[11735]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1805
(book
6.214)
Date
solved
:
Wednesday, March 05, 2025 at 02:45:44 PM
CAS
classification
:
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\begin{align*} \left (4 y^{3}-a y-b \right ) y^{\prime \prime }-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2}&=0 \end{align*}
✓ Maple. Time used: 0.117 (sec). Leaf size: 250
ode:=(4*y(x)^3-a*y(x)-b)*diff(diff(y(x),x),x)-(6*y(x)^2-1/2*a)*diff(y(x),x)^2 = 0;
dsolve(ode,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}}} \\
\int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}-\textit {\_a} a -b}}d \textit {\_a} -c_{1} x -c_{2} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 11.862 (sec). Leaf size: 416
ode=(a/2 - 6*y[x]^2)*D[y[x],x]^2 + (-b - a*y[x] + 4*y[x]^3)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt {2} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}} \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-y(x)}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]}}\right ),\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,1\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}\right )}{c_1 \sqrt {a y(x)+b-4 y(x)^3} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}}}=x+c_2,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq((a/2 - 6*y(x)**2)*Derivative(y(x), x)**2 + (-a*y(x) - b + 4*y(x)**3)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(2)*sqrt((a*y(x) + b - 4*y(x)**3)*Derivative(y(x), (x, 2))/(a - 12*y(x)**2)) + Derivative(y(x), x) cannot be solved by the factorable group method