77.1.91 problem 118 (page 177)
Internal
problem
ID
[17910]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
118
(page
177)
Date
solved
:
Monday, March 31, 2025 at 04:49:51 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{2} y^{\prime \prime }-3 x y^{2} y^{\prime }+4 y^{3}+x^{6}&=0 \end{align*}
✓ Maple. Time used: 0.145 (sec). Leaf size: 112
ode:=x^2*y(x)^2*diff(diff(y(x),x),x)-3*x*y(x)^2*diff(y(x),x)+4*y(x)^3+x^6 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+{\cos \left (\operatorname {RootOf}\left (-\textit {\_Z} -\operatorname {csgn}\left (c_1 \right ) \ln \left (x \right ) c_1^{3}+\operatorname {csgn}\left (c_1 \right ) c_1^{3} c_2 +\operatorname {csgn}\left (c_1 \right ) \sqrt {\frac {\cos \left (\textit {\_Z} \right )^{2}}{c_1^{2}}}\, c_1 \right )\right )}^{2}-2 \textit {\_Z} \right )}{c_1^{2}} \\
y &= \frac {x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+{\cos \left (\operatorname {RootOf}\left (\textit {\_Z} -\operatorname {csgn}\left (c_1 \right ) \ln \left (x \right ) c_1^{3}+\operatorname {csgn}\left (c_1 \right ) c_1^{3} c_2 -\operatorname {csgn}\left (c_1 \right ) \sqrt {\frac {\cos \left (\textit {\_Z} \right )^{2}}{c_1^{2}}}\, c_1 \right )\right )}^{2}-2 \textit {\_Z} \right )}{c_1^{2}} \\
\end{align*}
✓ Mathematica. Time used: 3.04 (sec). Leaf size: 266
ode=x^2*y[x]^2*D[y[x],{x,2}]-3*x*y[x]^2*D[y[x],x]+4*y[x]^3+x^6==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
\text {Solve}\left [\log (x)-\frac {-2 x^3 \sqrt {2+\frac {c_1 y(x)}{x^2}} \text {arcsinh}\left (\frac {\sqrt {c_1} \sqrt {y(x)}}{\sqrt {2} x}\right )+2 \sqrt {c_1} x^2 \sqrt {y(x)}+c_1{}^{3/2} y(x)^{3/2}}{c_1{}^{3/2} x^2 \sqrt {-\frac {x^4}{y(x)^2}} \sqrt {y(x)} \sqrt {-\frac {y(x) \left (2 x^2+c_1 y(x)\right )}{x^4}}}&=c_2,y(x)\right ] \\
\text {Solve}\left [\log (x)+\frac {-2 x^3 \sqrt {2+\frac {c_1 y(x)}{x^2}} \text {arcsinh}\left (\frac {\sqrt {c_1} \sqrt {y(x)}}{\sqrt {2} x}\right )+2 \sqrt {c_1} x^2 \sqrt {y(x)}+c_1{}^{3/2} y(x)^{3/2}}{c_1{}^{3/2} x^2 \sqrt {-\frac {x^4}{y(x)^2}} \sqrt {y(x)} \sqrt {-\frac {y(x) \left (2 x^2+c_1 y(x)\right )}{x^4}}}&=c_2,y(x)\right ] \\
\end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**6 + x**2*y(x)**2*Derivative(y(x), (x, 2)) - 3*x*y(x)**2*Derivative(y(x), x) + 4*y(x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -x**5/(3*y(x)**2) - x*Derivative(y(x), (x, 2))/3 + Derivative(y(x), x) - 4*y(x)/(3*x) cannot be solved by the factorable group method