74.1.9 problem 15
Internal
problem
ID
[15789]
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
:
Tuesday, January 28, 2025 at 08:07:14 AM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} 2 x -y-y^{\prime } y&=0 \end{align*}
✓ Solution by Maple
Time used: 1.958 (sec). Leaf size: 1029
dsolve((2*x-y(x))-y(x)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= x \left (-1+\frac {\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}}{2 c_{1} x}+\frac {2 c_{1} x}{\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}}\right ) \\
y &= \frac {4 c_{1}^{2} x^{2}-2 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}+\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}}{2 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}} \\
y &= \frac {4 c_{1}^{2} x^{2}-2 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}+\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}}{2 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}} \\
y &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}-4 x c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}\right )}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}} c_{1}} \\
y &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}-4 x c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}\right )}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}} c_{1}} \\
y &= \frac {4 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}-4 c_{1}^{2} x^{2}-4 c_{1} x \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}-\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}}{4 c_{1} \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{2}/{3}}-4 x c_{1} \left (i x c_{1} \sqrt {3}+c_{1} x +\left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}}\right )}{4 \left (8 c_{1}^{3} x^{3}+4 \sqrt {4 c_{1}^{3} x^{3}+1}+4\right )^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 38
DSolve[(2*x-y[x])-y[x]*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {K[1]}{(K[1]-1) (K[1]+2)}dK[1]=-\log (x)+c_1,y(x)\right ]
\]