83.10.8 problem 8
Internal
problem
ID
[21977]
Book
:
Differential
Equations
By
Kaj
L.
Nielsen.
Second
edition
1966.
Barnes
and
nobel.
66-28306
Section
:
Chapter
IV.
First
order
differential
equations
of
higher
degree.
Ex.
XI
at
page
69
Problem
number
:
8
Date
solved
:
Thursday, October 02, 2025 at 08:20:59 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} x +y {y^{\prime }}^{2}&=0 \end{align*}
✓ Maple. Time used: 0.012 (sec). Leaf size: 101
ode:=x+y(x)*diff(y(x),x)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
x +\frac {x c_1}{\left (\frac {-\sqrt {-y x}+y}{y}\right )^{{2}/{3}} \left (\frac {-x +\sqrt {-y x}+y}{y}\right )^{{2}/{3}} y} &= 0 \\
x +\frac {x c_1}{\left (\frac {\sqrt {-y x}+y}{y}\right )^{{2}/{3}} \left (\frac {-x -\sqrt {-y x}+y}{y}\right )^{{2}/{3}} y} &= 0 \\
\end{align*}
✓ Mathematica. Time used: 3.381 (sec). Leaf size: 49
ode=x+y[x]*D[y[x],x]^2==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \left (\frac {3 c_1}{2}-i x^{3/2}\right ){}^{2/3}\\ y(x)&\to \left (i x^{3/2}+\frac {3 c_1}{2}\right ){}^{2/3} \end{align*}
✓ Sympy. Time used: 30.560 (sec). Leaf size: 374
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x + y(x)*Derivative(y(x), x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} - 2 C_{1} \sqrt {- x^{3}} - x^{3}}, \ y{\left (x \right )} = \sqrt [3]{C_{1}^{2} + 2 C_{1} \sqrt {- x^{3}} - x^{3}}\right ]
\]