83.43.18 problem Ex 19 page 23
Internal
problem
ID
[19526]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Book
Solved
Excercises.
Chapter
II.
Equations
of
first
order
and
first
degree
Problem
number
:
Ex
19
page
23
Date
solved
:
Tuesday, January 28, 2025 at 01:49:21 PM
CAS
classification
:
[_rational]
\begin{align*} y+\frac {y^{3}}{3}+\frac {x^{2}}{2}+\frac {\left (x +x y^{2}\right ) y^{\prime }}{4}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 313
dsolve((y(x)+1/3*y(x)^3+1/2*x^2)+1/4*(x+x*y(x)^2 )*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {2^{{1}/{3}} \left (x^{4}-\frac {2^{{1}/{3}} {\left (\left (-x^{6}+\sqrt {x^{12}+4 x^{8}+24 c_{1} x^{6}+144 c_{1}^{2}}-12 c_{1} \right ) x^{2}\right )}^{{2}/{3}}}{2}\right )}{{\left (\left (-x^{6}+\sqrt {x^{12}+4 x^{8}+24 c_{1} x^{6}+144 c_{1}^{2}}-12 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\
y \left (x \right ) &= -\frac {\left (2^{{1}/{3}} \left (1+i \sqrt {3}\right ) {\left (\left (-x^{6}+\sqrt {x^{12}+4 x^{8}+24 c_{1} x^{6}+144 c_{1}^{2}}-12 c_{1} \right ) x^{2}\right )}^{{2}/{3}}+2 x^{4} \left (i \sqrt {3}-1\right )\right ) 2^{{1}/{3}}}{4 {\left (\left (-x^{6}+\sqrt {x^{12}+4 x^{8}+24 c_{1} x^{6}+144 c_{1}^{2}}-12 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\
y \left (x \right ) &= \frac {2^{{1}/{3}} \left (2^{{1}/{3}} \left (i \sqrt {3}-1\right ) {\left (\left (-x^{6}+\sqrt {x^{12}+4 x^{8}+24 c_{1} x^{6}+144 c_{1}^{2}}-12 c_{1} \right ) x^{2}\right )}^{{2}/{3}}+2 x^{4} \left (1+i \sqrt {3}\right )\right )}{4 {\left (\left (-x^{6}+\sqrt {x^{12}+4 x^{8}+24 c_{1} x^{6}+144 c_{1}^{2}}-12 c_{1} \right ) x^{2}\right )}^{{1}/{3}} x^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.108 (sec). Leaf size: 393
DSolve[(y[x]+1/3*y[x]^3+1/2*x^2)+1/4*(x+x*y[x]^2 )*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-2 \sqrt [3]{2} x^8+2^{2/3} \left (-x^{14}+c_1 x^8+\sqrt {x^{16} \left (x^{12}+4 x^8-2 c_1 x^6+c_1{}^2\right )}\right ){}^{2/3}}{2 x^4 \sqrt [3]{-x^{14}+c_1 x^8+\sqrt {x^{16} \left (x^{12}+4 x^8-2 c_1 x^6+c_1{}^2\right )}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^8+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-x^{14}+c_1 x^8+\sqrt {x^{16} \left (x^{12}+4 x^8-2 c_1 x^6+c_1{}^2\right )}\right ){}^{2/3}}{4 x^4 \sqrt [3]{-x^{14}+c_1 x^8+\sqrt {x^{16} \left (x^{12}+4 x^8-2 c_1 x^6+c_1{}^2\right )}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x^8-i 2^{2/3} \left (\sqrt {3}-i\right ) \left (-x^{14}+c_1 x^8+\sqrt {x^{16} \left (x^{12}+4 x^8-2 c_1 x^6+c_1{}^2\right )}\right ){}^{2/3}}{4 x^4 \sqrt [3]{-x^{14}+c_1 x^8+\sqrt {x^{16} \left (x^{12}+4 x^8-2 c_1 x^6+c_1{}^2\right )}}} \\
\end{align*}