77.1.91 problem 118 (page 177)
Internal
problem
ID
[17981]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
118
(page
177)
Date
solved
:
Tuesday, January 28, 2025 at 08:28:24 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*}
✓ Solution by Maple
Time used: 0.368 (sec). Leaf size: 112
dsolve(x^2*y(x)^2*diff(y(x),x$2)-3*x*y(x)^2*diff(y(x),x)+4*y(x)^3+x^6=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\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 ) x^{2}}{c_{1}^{2}} \\
y &= \frac {\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 ) x^{2}}{c_{1}^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 3.040 (sec). Leaf size: 266
DSolve[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,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*}