77.23.13 problem 13
Internal
problem
ID
[20579]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
V.
Singular
solutions.
Exercise
V
at
page
76
Problem
number
:
13
Date
solved
:
Thursday, October 02, 2025 at 06:10:08 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y {y^{\prime }}^{2}-2 x y^{\prime }+y&=0 \end{align*}
✓ Maple. Time used: 0.341 (sec). Leaf size: 67
ode:=y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -x \\
y &= x \\
y &= 0 \\
y &= \sqrt {c_1 \left (-2 i x +c_1 \right )} \\
y &= \sqrt {c_1 \left (2 i x +c_1 \right )} \\
y &= -\sqrt {c_1 \left (-2 i x +c_1 \right )} \\
y &= -\sqrt {c_1 \left (2 i x +c_1 \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.823 (sec). Leaf size: 64
ode=y[x]*D[y[x],x]^2-2*x*D[y[x],x]+y[x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )}\\ y(x)&\to \sqrt {-e^{c_1} \left (-2 x+e^{c_1}\right )}\\ y(x)&\to 0\\ y(x)&\to -x\\ y(x)&\to x \end{align*}
✓ Sympy. Time used: 148.816 (sec). Leaf size: 469
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*x*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x - C_{1}}}{2}\right ]
\]