77.21.10 problem 10
Internal
problem
ID
[20539]
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
(D)
at
page
57
Problem
number
:
10
Date
solved
:
Thursday, October 02, 2025 at 06:04:41 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} 4 y {y^{\prime }}^{2}+2 x y^{\prime }-y&=0 \end{align*}
✓ Maple. Time used: 0.059 (sec). Leaf size: 65
ode:=4*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 &= -\frac {i x}{2} \\
y &= \frac {i x}{2} \\
y &= 0 \\
y &= \sqrt {c_1 \left (c_1 -x \right )} \\
y &= \sqrt {c_1 \left (c_1 +x \right )} \\
y &= -\sqrt {c_1 \left (c_1 -x \right )} \\
y &= -\sqrt {c_1 \left (c_1 +x \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.345 (sec). Leaf size: 140
ode=4*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 -\frac {1}{2} e^{2 c_1} \sqrt {-2 x+e^{4 c_1}}\\ y(x)&\to \frac {1}{2} e^{2 c_1} \sqrt {-2 x+e^{4 c_1}}\\ y(x)&\to -\frac {1}{2} e^{2 c_1} \sqrt {2 x+e^{4 c_1}}\\ y(x)&\to \frac {1}{2} e^{2 c_1} \sqrt {2 x+e^{4 c_1}}\\ y(x)&\to 0\\ y(x)&\to -\frac {i x}{2}\\ y(x)&\to \frac {i x}{2} \end{align*}
✓ Sympy. Time used: 142.617 (sec). Leaf size: 469
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*Derivative(y(x), x) + 4*y(x)*Derivative(y(x), x)**2 - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{4}\right ]
\]