83.43.18 problem Ex 19 page 23
Internal
problem
ID
[19455]
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
:
Monday, March 31, 2025 at 07:18:29 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*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 313
ode:=y(x)+1/3*y(x)^3+1/2*x^2+1/4*(x+x*y(x)^2)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\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 &= -\frac {\left (\left (1+i \sqrt {3}\right ) 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 \left (i \sqrt {3}-1\right ) x^{4}\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 &= \frac {\left (\left (i \sqrt {3}-1\right ) 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 \left (1+i \sqrt {3}\right ) x^{4}\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}} \\
\end{align*}
✓ Mathematica. Time used: 60.108 (sec). Leaf size: 393
ode=(y[x]+1/3*y[x]^3+1/2*x^2)+1/4*(x+x*y[x]^2 )*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},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*}
✓ Sympy. Time used: 33.746 (sec). Leaf size: 267
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2/2 + (x*y(x)**2 + x)*Derivative(y(x), x)/4 + y(x)**3/3 + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \left (-1 - \sqrt {3} i\right ) \sqrt [3]{\sqrt {4 + \frac {\left (3 C_{1} + x^{6}\right )^{2}}{x^{8}}} + \frac {3 C_{1} + x^{6}}{x^{4}}}}{4} + \frac {2}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{\sqrt {4 + \frac {\left (3 C_{1} + x^{6}\right )^{2}}{x^{8}}} + \frac {3 C_{1} + x^{6}}{x^{4}}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \left (-1 + \sqrt {3} i\right ) \sqrt [3]{\sqrt {4 + \frac {\left (3 C_{1} + x^{6}\right )^{2}}{x^{8}}} + \frac {3 C_{1} + x^{6}}{x^{4}}}}{4} + \frac {2}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{\sqrt {4 + \frac {\left (3 C_{1} + x^{6}\right )^{2}}{x^{8}}} + \frac {3 C_{1} + x^{6}}{x^{4}}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \sqrt [3]{\sqrt {4 + \frac {\left (3 C_{1} + x^{6}\right )^{2}}{x^{8}}} + \frac {3 C_{1} + x^{6}}{x^{4}}}}{2} + \frac {1}{\sqrt [3]{\sqrt {4 + \frac {\left (3 C_{1} + x^{6}\right )^{2}}{x^{8}}} + \frac {3 C_{1} + x^{6}}{x^{4}}}}\right )\right ]
\]