68.1.9 problem 15
Internal
problem
ID
[17076]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
1.
Introduction
to
Differential
Equations.
Exercises
1.1,
page
10
Problem
number
:
15
Date
solved
:
Thursday, October 02, 2025 at 01:42:40 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} 2 x -y-y y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.121 (sec). Leaf size: 1224
ode:=2*x-y(x)-y(x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 44.868 (sec). Leaf size: 496
ode=(2*x-y[x])-y[x]*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-x\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-x\\ y(x)&\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{2^{2/3} \sqrt [3]{2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}-e^{3 c_1}}}-x\\ y(x)&\to \sqrt [3]{x^3}+\frac {\left (x^3\right )^{2/3}}{x}-x\\ y(x)&\to \frac {1}{2} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{x^3}+\frac {\left (-1-i \sqrt {3}\right ) \left (x^3\right )^{2/3}}{x}-2 x\right )\\ y(x)&\to \frac {1}{2} \left (\left (-1-i \sqrt {3}\right ) \sqrt [3]{x^3}+\frac {i \left (\sqrt {3}+i\right ) \left (x^3\right )^{2/3}}{x}-2 x\right ) \end{align*}
✓ Sympy. Time used: 16.986 (sec). Leaf size: 250
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x - y(x)*Derivative(y(x), x) - y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {2 x^{2}}{\sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}} - x + \sqrt {3} i x - \sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}} - \sqrt {3} i \sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}}{1 - \sqrt {3} i}, \ y{\left (x \right )} = \frac {\frac {2 x^{2}}{\sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}} - x - \sqrt {3} i x - \sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}} + \sqrt {3} i \sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}}{1 + \sqrt {3} i}, \ y{\left (x \right )} = - \frac {x^{2}}{\sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}} - x - \sqrt [3]{2 C_{1} - x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}\right ]
\]