83.7.7 problem 5 (a)
Internal
problem
ID
[19125]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
II.
Equations
of
first
order
and
first
degree.
Exercise
II
(F)
at
page
24
Problem
number
:
5
(a)
Date
solved
:
Tuesday, January 28, 2025 at 12:58:49 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{2}+2 x^{2} y+\left (2 x^{3}-y x \right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.269 (sec). Leaf size: 1196
dsolve((y(x)^2+2*x^2*y(x))+(2*x^3-x*y(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {4 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{\left (16 x^{2}-4 x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}\right )^{2}} \\
y \left (x \right ) &= \frac {4 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{\left (16 x^{2}-4 x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}\right )^{2}} \\
y \left (x \right ) &= -\frac {256 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{\left (16 x^{2}-4 x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}\right )^{2} \left (\sqrt {3}+i\right )^{6}} \\
y \left (x \right ) &= \frac {16 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{{\left (4 x +\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}\right )}^{2} {\left (i \left (\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}-4 x \right ) \sqrt {3}+4 x +\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}\right )}^{2}} \\
y \left (x \right ) &= \frac {16 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{{\left (4 x +\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}\right )}^{2} {\left (-i \left (\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}-4 x \right ) \sqrt {3}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+4 x \right )}^{2}} \\
y \left (x \right ) &= \frac {16 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{{\left (4 x +\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}\right )}^{2} {\left (i \left (\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}-4 x \right ) \sqrt {3}+4 x +\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}\right )}^{2}} \\
y \left (x \right ) &= \frac {16 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{{\left (4 x +\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}\right )}^{2} {\left (-i \left (\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}-4 x \right ) \sqrt {3}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+4 x \right )}^{2}} \\
y \left (x \right ) &= \frac {1024 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{\left (i \sqrt {3}-1\right )^{6} {\left (i \sqrt {3}\, \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}-16 i \sqrt {3}\, x^{2}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}+8 x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+16 x^{2}\right )}^{2}} \\
y \left (x \right ) &= \frac {1024 c_{1}^{3} x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}}{\left (i \sqrt {3}-1\right )^{6} \left (16 i \sqrt {3}\, x^{2}-i \sqrt {3}\, \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}+16 x^{2}+8 x \left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{1}/{3}}+\left (4 c_{1}^{3}-64 x^{3}+4 c_{1} \sqrt {c_{1}^{4}-32 c_{1} x^{3}}\right )^{{2}/{3}}\right )^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 47.865 (sec). Leaf size: 546
DSolve[(y[x]^2+2*x^2*y[x])+(2*x^3-x*y[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to 4 x^2+\frac {\sqrt [3]{-16 x^6+e^{3 c_1} x^3+\sqrt {e^{3 c_1} x^6 \left (-32 x^3+e^{3 c_1}\right )}}}{\sqrt [3]{2}}+\frac {4 \sqrt [3]{2} x^4}{\sqrt [3]{-16 x^6+e^{3 c_1} x^3+\sqrt {e^{3 c_1} x^6 \left (-32 x^3+e^{3 c_1}\right )}}} \\
y(x)\to 4 x^2+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-16 x^6+e^{3 c_1} x^3+\sqrt {e^{3 c_1} x^6 \left (-32 x^3+e^{3 c_1}\right )}}}{2 \sqrt [3]{2}}-\frac {2 \sqrt [3]{2} \left (1+i \sqrt {3}\right ) x^4}{\sqrt [3]{-16 x^6+e^{3 c_1} x^3+\sqrt {e^{3 c_1} x^6 \left (-32 x^3+e^{3 c_1}\right )}}} \\
y(x)\to 4 x^2-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-16 x^6+e^{3 c_1} x^3+\sqrt {e^{3 c_1} x^6 \left (-32 x^3+e^{3 c_1}\right )}}}{2 \sqrt [3]{2}}+\frac {2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) x^4}{\sqrt [3]{-16 x^6+e^{3 c_1} x^3+\sqrt {e^{3 c_1} x^6 \left (-32 x^3+e^{3 c_1}\right )}}} \\
y(x)\to \frac {2 \left (\sqrt [3]{-x^6}+x^2\right )^2}{\sqrt [3]{-x^6}} \\
y(x)\to \left (-1-i \sqrt {3}\right ) \sqrt [3]{-x^6}+4 x^2+\frac {\left (1-i \sqrt {3}\right ) \left (-x^6\right )^{2/3}}{x^2} \\
y(x)\to i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^6}+4 x^2+\frac {\left (1+i \sqrt {3}\right ) \left (-x^6\right )^{2/3}}{x^2} \\
\end{align*}