60.7.186 problem 1806 (book 6.215)
Internal
problem
ID
[11736]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1806
(book
6.215)
Date
solved
:
Sunday, March 30, 2025 at 09:05:16 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 ) \left (y^{\prime \prime }+f y^{\prime }\right )-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2}&=0 \end{align*}
✓ Maple. Time used: 0.052 (sec). Leaf size: 253
ode:=(4*y(x)^3-a*y(x)-b)*(diff(diff(y(x),x),x)+f*diff(y(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}}} \\
c_1 \,{\mathrm e}^{-f x}-c_2 +\int _{}^{y}\frac {1}{\sqrt {4 \textit {\_a}^{3}-\textit {\_a} a -b}}d \textit {\_a} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 10.275 (sec). Leaf size: 438
ode=(a/2 - 6*y[x]^2)*D[y[x],x]^2 + (-b - a*y[x] + 4*y[x]^3)*(f[x]*D[y[x],x] + D[y[x],{x,2}]) == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {2 \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}} \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-y(x)}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]}}\right ),\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,1\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}\right )}{\sqrt {a y(x)+b-4 y(x)^3} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}}}=\int _1^x-\sqrt {2} \exp \left (-\int _1^{K[1]}f(K[1])dK[1]\right ) c_1dK[1]+c_2,y(x)\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
f = symbols("f")
y = Function("y")
ode = Eq((a/2 - 6*y(x)**2)*Derivative(y(x), x)**2 + (f*Derivative(y(x), x) + Derivative(y(x), (x, 2)))*(-a*y(x) - b + 4*y(x)**3),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (f*(a*y(x) + b - 4*y(x)**3) + sqrt((a*y(x) + b - 4*y(x)**3)*(a*f**2*y(x) + 2*a*Derivative(y(x), (x, 2)) + b*f**2 - 4*f**2*y(x)**3 - 24*y(x)**2*Derivative(y(x), (x, 2)))))/(a - 12*y(x)**2) cannot be solved by the factorable group method