81.3.24 problem 24
Internal
problem
ID
[18642]
Book
:
A
short
course
on
differential
equations.
By
Donald
Francis
Campbell.
Maxmillan
company.
London.
1907
Section
:
Chapter
III.
Ordinary
differential
equations
of
the
first
order
and
first
degree.
Exercises
at
page
33
Problem
number
:
24
Date
solved
:
Tuesday, January 28, 2025 at 12:07:05 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (x^{2}+3 y x -y^{2}\right ) y^{\prime }-3 y^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.717 (sec). Leaf size: 98
dsolve((x^2+3*x*y(x)-y(x)^2)*diff(y(x),x)-(3*y(x)^2)=0,y(x), singsol=all)
\[
y \left (x \right ) = \frac {x \left (-\operatorname {RootOf}\left (8 \textit {\_Z}^{10}-12 \textit {\_Z}^{8}+6 \textit {\_Z}^{6}+\left (c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-2 x^{2} c_{1} \textit {\_Z}^{2}+c_{1} x^{2}\right )^{2}+1\right )}{\operatorname {RootOf}\left (8 \textit {\_Z}^{10}-12 \textit {\_Z}^{8}+6 \textit {\_Z}^{6}+\left (c_{1} x^{2}-1\right ) \textit {\_Z}^{4}-2 x^{2} c_{1} \textit {\_Z}^{2}+c_{1} x^{2}\right )^{2}}
\]
✓ Solution by Mathematica
Time used: 3.866 (sec). Leaf size: 396
DSolve[(x^2+3*x*y[x]-y[x]^2)*D[y[x],x]-(3*y[x]^2)==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,1\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,2\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,3\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,4\right ] \\
y(x)\to \text {Root}\left [\text {$\#$1}^5+3 \text {$\#$1}^4 x+\text {$\#$1}^3 \left (3 x^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (x^3-3 e^{2 c_1} x\right )+3 \text {$\#$1} e^{2 c_1} x^2-e^{2 c_1} x^3\&,5\right ] \\
y(x)\to 0 \\
\end{align*}