50.4.10 problem 10
Internal
problem
ID
[7859]
Book
:
Differential
Equations:
Theory,
Technique,
and
Practice
by
George
Simmons,
Steven
Krantz.
McGraw-Hill
NY.
2007.
1st
Edition.
Section
:
Chapter
1.
What
is
a
differential
equation.
Section
1.5.
Exact
Equations.
Page
20
Problem
number
:
10
Date
solved
:
Monday, January 27, 2025 at 03:27:12 PM
CAS
classification
:
[_exact, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} 2 x y^{3}+\cos \left (x \right ) y+\left (3 x^{2} y^{2}+\sin \left (x \right )\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 298
dsolve((2*x*y(x)^3+y(x)*cos(x))+(3*x^2*y(x)^2+sin(x))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y &= \frac {\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{2}/{3}}-12 \sin \left (x \right )}{6 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{1}/{3}}} \\
y &= -\frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{2}/{3}}+12 i \sin \left (x \right ) \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{2}/{3}}-12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{2}/{3}}+12 i \sin \left (x \right ) \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{2}/{3}}+12 \sin \left (x \right )}{12 x \left (12 \sqrt {3}\, \sqrt {27 x^{2} c_{1}^{2}+4 \sin \left (x \right )^{3}}-108 c_{1} x \right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 39.530 (sec). Leaf size: 339
DSolve[(2*x*y[x]^3+y[x]*Cos[x])+(3*x^2*y[x]^2+Sin[x])*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{9 c_1 x^4+\sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}}{\sqrt [3]{2} 3^{2/3} x^2}-\frac {\sqrt [3]{\frac {2}{3}} \sin (x)}{\sqrt [3]{9 c_1 x^4+\sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}} \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sin (x)}{2^{2/3} \sqrt [3]{27 c_1 x^4+3 \sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^4+\sqrt {108 x^6 \sin ^3(x)+729 c_1{}^2 x^8}}}{6 \sqrt [3]{2} x^2} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sin (x)}{2^{2/3} \sqrt [3]{27 c_1 x^4+3 \sqrt {12 x^6 \sin ^3(x)+81 c_1{}^2 x^8}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^4+\sqrt {108 x^6 \sin ^3(x)+729 c_1{}^2 x^8}}}{6 \sqrt [3]{2} x^2} \\
\end{align*}