60.7.140 problem 1756 (book 6.165)
Internal
problem
ID
[11690]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1756
(book
6.165)
Date
solved
:
Sunday, March 30, 2025 at 08:42:37 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} a y y^{\prime \prime }+b {y^{\prime }}^{2}+\operatorname {c4} y^{4}+\operatorname {c3} y^{3}+\operatorname {c2} y^{2}+\operatorname {c1} y+\operatorname {c0}&=0 \end{align*}
✓ Maple. Time used: 0.045 (sec). Leaf size: 419
ode:=a*y(x)*diff(diff(y(x),x),x)+b*diff(y(x),x)^2+c4*y(x)^4+c3*y(x)^3+c2*y(x)^2+c1*y(x)+c0 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
6 \left (a +b \right ) \left (a +\frac {2 b}{3}\right ) b \left (a +\frac {b}{2}\right ) \left (a +2 b \right ) \int _{}^{y}\frac {\textit {\_a}^{\frac {2 b}{a}}}{\sqrt {-36 \left (a +b \right ) \left (a +\frac {2 b}{3}\right ) \textit {\_a}^{\frac {2 b}{a}} b \left (a +\frac {b}{2}\right ) \left (\frac {2 \left (a +b \right ) \operatorname {c3} b \left (a +\frac {b}{2}\right ) \left (a +2 b \right ) \textit {\_a}^{\frac {3 a +2 b}{a}}}{3}+\left (a +\frac {2 b}{3}\right ) \left (\operatorname {c2} b \left (a +\frac {b}{2}\right ) \left (a +2 b \right ) \textit {\_a}^{\frac {2 a +2 b}{a}}+\left (a +b \right ) \left (\frac {b \operatorname {c4} \left (a +2 b \right ) \textit {\_a}^{\frac {4 a +2 b}{a}}}{2}+\left (a +\frac {b}{2}\right ) \left (2 \textit {\_a}^{\frac {a +2 b}{a}} b \operatorname {c1} +\left (\textit {\_a}^{\frac {2 b}{a}} \operatorname {c0} -c_1 b \right ) \left (a +2 b \right )\right )\right )\right )\right ) \left (a +2 b \right )}}d \textit {\_a} -c_2 -x &= 0 \\
-6 \left (a +b \right ) \left (a +\frac {2 b}{3}\right ) b \left (a +\frac {b}{2}\right ) \left (a +2 b \right ) \int _{}^{y}\frac {\textit {\_a}^{\frac {2 b}{a}}}{\sqrt {-36 \left (a +b \right ) \left (a +\frac {2 b}{3}\right ) \textit {\_a}^{\frac {2 b}{a}} b \left (a +\frac {b}{2}\right ) \left (\frac {2 \left (a +b \right ) \operatorname {c3} b \left (a +\frac {b}{2}\right ) \left (a +2 b \right ) \textit {\_a}^{\frac {3 a +2 b}{a}}}{3}+\left (a +\frac {2 b}{3}\right ) \left (\operatorname {c2} b \left (a +\frac {b}{2}\right ) \left (a +2 b \right ) \textit {\_a}^{\frac {2 a +2 b}{a}}+\left (a +b \right ) \left (\frac {b \operatorname {c4} \left (a +2 b \right ) \textit {\_a}^{\frac {4 a +2 b}{a}}}{2}+\left (a +\frac {b}{2}\right ) \left (2 \textit {\_a}^{\frac {a +2 b}{a}} b \operatorname {c1} +\left (\textit {\_a}^{\frac {2 b}{a}} \operatorname {c0} -c_1 b \right ) \left (a +2 b \right )\right )\right )\right )\right ) \left (a +2 b \right )}}d \textit {\_a} -c_2 -x &= 0 \\
\end{align*}
✓ Mathematica. Time used: 10.038 (sec). Leaf size: 2166
ode=c0 + c1*y[x] + c2*y[x]^2 + c3*y[x]^3 + c4*y[x]^4 + b*D[y[x],x]^2 + a*y[x]*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Too large to display
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c0 = symbols("c0")
c1 = symbols("c1")
c2 = symbols("c2")
c3 = symbols("c3")
c4 = symbols("c4")
y = Function("y")
ode = Eq(a*y(x)*Derivative(y(x), (x, 2)) + b*Derivative(y(x), x)**2 + c0 + c1*y(x) + c2*y(x)**2 + c3*y(x)**3 + c4*y(x)**4,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE sqrt(-(a*y(x)*Derivative(y(x), (x, 2)) + c0 + c1*y(x) + c2*y(x)**2 + c3*y(x)**3 + c4*y(x)**4)/b) + Derivative(y(x), x) cannot be solved by the factorable group method