77.1.85 problem 112 (page 172)
Internal
problem
ID
[17975]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
112
(page
172)
Date
solved
:
Tuesday, January 28, 2025 at 11:19:13 AM
CAS
classification
:
[[_3rd_order, _missing_y], [_3rd_order, _with_linear_symmetries]]
\begin{align*} y^{\prime \prime }-x y^{\prime \prime \prime }+{y^{\prime \prime \prime }}^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 9.738 (sec). Leaf size: 288
dsolve(diff(y(x),x$2)-x*diff(y(x),x$3)+diff(y(x),x$3)^3=0,y(x), singsol=all)
\begin{align*}
y &= -\frac {8 x^{{7}/{2}} \sqrt {3}}{315}+c_{1} x +c_{2} \\
y &= \frac {8 x^{{7}/{2}} \sqrt {3}}{315}+c_{1} x +c_{2} \\
y &= \frac {1}{6} c_{1} x^{3}-\frac {1}{2} c_{1}^{3} x^{2}+c_{2} x +c_{3} \\
y &= \int \left (\int \operatorname {RootOf}\left (-9 \ln \left (x \right )+2 \left (\int _{}^{\textit {\_Z}}-\frac {-27 \textit {\_f}^{2} 3^{{2}/{3}} 2^{{2}/{3}} {\left (\frac {\left (\sqrt {81 \textit {\_f}^{2}-12}+9 \textit {\_f} \right )^{2}}{\left (27 \textit {\_f}^{2}-4\right )^{3}}\right )}^{{1}/{3}}+81 {\left (\frac {\sqrt {81 \textit {\_f}^{2}-12}+9 \textit {\_f}}{\left (27 \textit {\_f}^{2}-4\right ) \sqrt {81 \textit {\_f}^{2}-12}}\right )}^{{1}/{3}} \textit {\_f}^{2}+4 \,3^{{2}/{3}} 2^{{2}/{3}} {\left (\frac {\left (\sqrt {81 \textit {\_f}^{2}-12}+9 \textit {\_f} \right )^{2}}{\left (27 \textit {\_f}^{2}-4\right )^{3}}\right )}^{{1}/{3}}+6 \,2^{{1}/{3}}-12 {\left (\frac {\sqrt {81 \textit {\_f}^{2}-12}+9 \textit {\_f}}{\left (27 \textit {\_f}^{2}-4\right ) \sqrt {81 \textit {\_f}^{2}-12}}\right )}^{{1}/{3}}}{\left (27 \textit {\_f}^{2}-4\right ) \textit {\_f} {\left (\frac {\sqrt {81 \textit {\_f}^{2}-12}+9 \textit {\_f}}{\left (27 \textit {\_f}^{2}-4\right ) \sqrt {81 \textit {\_f}^{2}-12}}\right )}^{{1}/{3}}}d \textit {\_f} \right )+9 c_{1} \right ) x^{{3}/{2}}d x \right )d x +c_{2} x +c_{3} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 32
DSolve[D[y[x],{x,2}]-x*D[y[x],{x,3}]+D[y[x],{x,3}]^3==0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to \frac {c_1 x^3}{6}-\frac {c_1{}^3 x^2}{2}+c_3 x+c_2
\]