82.12.21 problem Ex. 23
Internal
problem
ID
[18786]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
II.
Equations
of
the
first
order
and
of
the
first
degree.
Examples
on
chapter
II
at
page
29
Problem
number
:
Ex.
23
Date
solved
:
Tuesday, January 28, 2025 at 12:17:22 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} \left (1+6 y^{2}-3 x^{2} y\right ) y^{\prime }&=3 x y^{2}-x^{2} \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 769
dsolve((1+6*y(x)^2-3*x^2*y(x))*diff(y(x),x)=3*x*y(x)^2-x^2,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 c_{1} x^{3}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}}{12}+\frac {3 x^{4}-8}{4 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 c_{1} x^{3}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}}+\frac {x^{2}}{4} \\
y \left (x \right ) &= \frac {24+i \left (-24+9 x^{4}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}}{24 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {24+i \left (-9 x^{4}+\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}+24\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{2}/{3}}}{24 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 7.878 (sec). Leaf size: 570
DSolve[(1+6*y[x]^2-3*x^2*y[x])*D[y[x],x]==3*x*y[x]^2-x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {x^2}{4}-\frac {\sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{6 \sqrt [3]{2}}+\frac {6-\frac {9 x^4}{4}}{3\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\
y(x)\to \frac {x^2}{4}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\
y(x)\to \frac {x^2}{4}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}}{12 \sqrt [3]{2}}-\frac {\left (1-i \sqrt {3}\right ) \left (6-\frac {9 x^4}{4}\right )}{6\ 2^{2/3} \sqrt [3]{-\frac {27 x^6}{4}+36 x^3+27 x^2+\sqrt {4 \left (6-\frac {9 x^4}{4}\right )^3+\left (-\frac {27 x^6}{4}+36 x^3+27 x^2+108 c_1\right ){}^2}+108 c_1}} \\
\end{align*}