83.22.5 problem 5
Internal
problem
ID
[19259]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
5
Date
solved
:
Tuesday, January 28, 2025 at 01:16:27 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} x +y^{\prime } y \left (2 {y^{\prime }}^{2}+3\right )&=0 \end{align*}
✓ Solution by Maple
Time used: 0.098 (sec). Leaf size: 768
dsolve(x+diff(y(x),x)*y(x)*(2*diff(y(x),x)^2+3)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {i \sqrt {2}\, x}{2} \\
y \left (x \right ) &= \frac {i \sqrt {2}\, x}{2} \\
y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {-2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}} \textit {\_a}^{2}+2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}} \textit {\_a}^{3}-{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}}+\textit {\_a} {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}}+\textit {\_a}^{2}}{{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +c_{1} \right ) x \\
y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}} \textit {\_a}^{2}+i \sqrt {3}\, {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}}+i \sqrt {3}\, \textit {\_a}^{2}-2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}} \textit {\_a}^{2}-4 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}} \textit {\_a}^{3}-{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}}-2 \textit {\_a} {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}}+\textit {\_a}^{2}}{{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\
y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}} \textit {\_a}^{2}+i \sqrt {3}\, {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}}+i \sqrt {3}\, \textit {\_a}^{2}+2 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}} \textit {\_a}^{2}+4 {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}} \textit {\_a}^{3}+{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{2}/{3}}+2 \textit {\_a} {\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}}-\textit {\_a}^{2}}{{\left (\frac {\left (\textit {\_a}^{2}-\sqrt {2 \textit {\_a}^{2}+1}+1\right ) \textit {\_a}}{\left (2 \textit {\_a}^{2}+1\right )^{{3}/{2}}}\right )}^{{1}/{3}} \left (2 \textit {\_a}^{4}+3 \textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\
\end{align*}
✗ Solution by Mathematica
Time used: 0.000 (sec). Leaf size: 0
DSolve[x+D[y[x],x]*y[x]*(2*D[y[x],x]^2+3)==0,y[x],x,IncludeSingularSolutions -> True]
Timed out