60.7.135 problem 1750 (book 6.159)
Internal
problem
ID
[11685]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1750
(book
6.159)
Date
solved
:
Wednesday, March 05, 2025 at 02:40:19 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} 4 y^{\prime \prime } y-3 {y^{\prime }}^{2}-12 y^{3}&=0 \end{align*}
✓ Maple. Time used: 0.108 (sec). Leaf size: 61
ode:=4*diff(diff(y(x),x),x)*y(x)-3*diff(y(x),x)^2-12*y(x)^3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
\int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{{3}/{2}} \left (4 \textit {\_a}^{{3}/{2}}+c_{1} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\
-\int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{{3}/{2}} \left (4 \textit {\_a}^{{3}/{2}}+c_{1} \right )}}d \textit {\_a} -x -c_{2} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.706 (sec). Leaf size: 469
ode=-12*y[x]^3 - 3*D[y[x],x]^2 + 4*y[x]*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [-\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \text {$\#$1} \sqrt {1-\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-4 \text {$\#$1}^{3/2}}{-c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}-c_1\right )}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {4 \text {$\#$1} \sqrt {1-\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},\frac {-4 \text {$\#$1}^{3/2}}{-c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}-c_1\right )}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [-\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\frac {4 \text {$\#$1} \sqrt {1+\frac {4 \text {$\#$1}^{3/2}}{c_1}} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{2},\frac {7}{6},-\frac {4 \text {$\#$1}^{3/2}}{c_1}\right )}{\sqrt {\text {$\#$1}^{3/2} \left (4 \text {$\#$1}^{3/2}+c_1\right )}}\&\right ][x+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-12*y(x)**3 + 4*y(x)*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -2*sqrt(3)*sqrt((-3*y(x)**2 + Derivative(y(x), (x, 2)))*y(x))/3 + Derivative(y(x), x) cannot be solved by the factorable group method