83.20.1 problem 1
Internal
problem
ID
[19239]
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
(C)
at
page
56
Problem
number
:
1
Date
solved
:
Tuesday, January 28, 2025 at 01:15:21 PM
CAS
classification
:
[_dAlembert]
\begin{align*} x&=y^{\prime } y-{y^{\prime }}^{2} \end{align*}
✓ Solution by Maple
Time used: 0.007 (sec). Leaf size: 299
dsolve(x=diff(y(x),x)*y(x)-diff(y(x),x)^2,y(x), singsol=all)
\begin{align*}
\frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}-4 x}\right ) c_{1}}{\sqrt {2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}-4 x}+4}\, \sqrt {-4+2 y \left (x \right )-2 \sqrt {y \left (x \right )^{2}-4 x}}}+x +\frac {\left (-y \left (x \right )+\sqrt {y \left (x \right )^{2}-4 x}\right ) \left (-\ln \left (2\right )+\ln \left (y \left (x \right )-\sqrt {y \left (x \right )^{2}-4 x}+\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}-4 x}-4 x -4}\right )\right )}{\sqrt {2 y \left (x \right )^{2}-2 y \left (x \right ) \sqrt {y \left (x \right )^{2}-4 x}-4 x -4}} &= 0 \\
\frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-4 x}\right ) c_{1}}{\sqrt {2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}-4 x}+4}\, \sqrt {-4+2 y \left (x \right )+2 \sqrt {y \left (x \right )^{2}-4 x}}}+x +\frac {\left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-4 x}\right ) \left (\ln \left (2\right )-\ln \left (y \left (x \right )+\sqrt {y \left (x \right )^{2}-4 x}+\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}-4 x}-4 x -4}\right )\right )}{\sqrt {2 y \left (x \right )^{2}+2 y \left (x \right ) \sqrt {y \left (x \right )^{2}-4 x}-4 x -4}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.627 (sec). Leaf size: 58
DSolve[x==D[y[x],x]*y[x]-D[y[x],x]^2,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{x=-\frac {K[1] \arcsin (K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}+K[1]\right \},\{y(x),K[1]\}\right ]
\]