60.7.172 problem 1792 (book 6.201)
Internal
problem
ID
[11722]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
6,
non-linear
second
order
Problem
number
:
1792
(book
6.201)
Date
solved
:
Sunday, March 30, 2025 at 08:44:39 PM
CAS
classification
:
[[_2nd_order, _missing_x]]
\begin{align*} \left (1-y\right ) y^{\prime \prime }-3 \left (1-2 y\right ) {y^{\prime }}^{2}-h \left (y\right )&=0 \end{align*}
✓ Maple. Time used: 0.084 (sec). Leaf size: 91
ode:=(1-y(x))*diff(diff(y(x),x),x)-3*(1-2*y(x))*diff(y(x),x)^2-h(y(x)) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {{\mathrm e}^{-6 \textit {\_b}}}{\sqrt {-2 \int \frac {{\mathrm e}^{-12 \textit {\_b}} h \left (\textit {\_b} \right )}{\left (-1+\textit {\_b} \right )^{7}}d \textit {\_b} +c_1}\, \left (-1+\textit {\_b} \right )^{3}}d \textit {\_b} -x -c_2 &= 0 \\
-\int _{}^{y}\frac {{\mathrm e}^{-6 \textit {\_b}}}{\sqrt {-2 \int \frac {{\mathrm e}^{-12 \textit {\_b}} h \left (\textit {\_b} \right )}{\left (-1+\textit {\_b} \right )^{7}}d \textit {\_b} +c_1}\, \left (-1+\textit {\_b} \right )^{3}}d \textit {\_b} -x -c_2 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 0.718 (sec). Leaf size: 590
ode=-h[y[x]] - 3*(1 - 2*y[x])*D[y[x],x]^2 + (1 - 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 [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right ) h(K[2])}{K[2]-1}dK[2]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right ) h(K[2])}{K[2]-1}dK[2]}}dK[4]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right )}{\sqrt {2 \int _1^{K[3]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right ) h(K[2])}{K[2]-1}dK[2]-c_1}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}-\frac {\exp \left (-\int _1^{K[3]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[3]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right ) h(K[2])}{K[2]-1}dK[2]}}dK[3]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right )}{\sqrt {2 \int _1^{K[4]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right ) h(K[2])}{K[2]-1}dK[2]-c_1}}dK[4]\&\right ][x+c_2] \\
y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\exp \left (-\int _1^{K[4]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right )}{\sqrt {c_1+2 \int _1^{K[4]}-\frac {\exp \left (-2 \int _1^{K[2]}\frac {6 K[1]-3}{K[1]-1}dK[1]\right ) h(K[2])}{K[2]-1}dK[2]}}dK[4]\&\right ][x+c_2] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
h = Function("h")
ode = Eq((1 - y(x))*Derivative(y(x), (x, 2)) - (3 - 6*y(x))*Derivative(y(x), x)**2 - h(y(x)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(3)*sqrt((h(y(x)) + y(x)*Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 2)))/(2*y(x) - 1))/3 + Derivative(y(x), x) cannot be solved by the factorable group method