83.20.1 problem 1
Internal
problem
ID
[19160]
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
:
Thursday, March 13, 2025 at 01:46:28 PM
CAS
classification
:
[_dAlembert]
\begin{align*} x&=y^{\prime } y-{y^{\prime }}^{2} \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 299
ode:=x = diff(y(x),x)*y(x)-diff(y(x),x)^2;
dsolve(ode,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*}
✓ Mathematica. Time used: 0.627 (sec). Leaf size: 58
ode=x==D[y[x],x]*y[x]-D[y[x],x]^2;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left \{x=-\frac {K[1] \arcsin (K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}+K[1]\right \},\{y(x),K[1]\}\right ]
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x - y(x)*Derivative(y(x), x) + Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE -sqrt(-4*x + y(x)**2)/2 - y(x)/2 + Derivative(y(x), x) cannot be solved by the factorable group method