76.16.2 problem Ex. 2
Internal
problem
ID
[20101]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
III.
Equations
of
the
first
order
but
not
of
the
first
degree.
Problems
at
page
36
Problem
number
:
Ex.
2
Date
solved
:
Thursday, October 02, 2025 at 05:25:45 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} y&=y {y^{\prime }}^{2}+2 x y^{\prime } \end{align*}
✓ Maple. Time used: 0.062 (sec). Leaf size: 69
ode:=y(x) = y(x)*diff(y(x),x)^2+2*x*diff(y(x),x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -i x \\
y &= i x \\
y &= 0 \\
y &= \sqrt {c_1 \left (c_1 -2 x \right )} \\
y &= \sqrt {c_1 \left (c_1 +2 x \right )} \\
y &= -\sqrt {c_1 \left (c_1 -2 x \right )} \\
y &= -\sqrt {c_1 \left (c_1 +2 x \right )} \\
\end{align*}
✓ Mathematica. Time used: 0.272 (sec). Leaf size: 126
ode=y[x]==y[x]*D[y[x],x]^2+2*D[y[x],x]*x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}}\\ y(x)&\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}}\\ y(x)&\to -e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}}\\ y(x)&\to e^{\frac {c_1}{2}} \sqrt {2 x+e^{c_1}}\\ y(x)&\to 0\\ y(x)&\to -i x\\ y(x)&\to i x \end{align*}
✓ Sympy. Time used: 141.390 (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)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} - 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- C_{1} + 2 \sqrt {2} x \sqrt {- C_{1}}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {2 \sqrt {2} \sqrt {C_{1}} x + C_{1}}}{2}\right ]
\]