40.2.26 problem 52
Internal
problem
ID
[6604]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
4.
Equations
of
first
order
and
first
degree
(Variable
separable).
Supplemetary
problems.
Page
22
Problem
number
:
52
Date
solved
:
Friday, March 14, 2025 at 01:48:53 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(y)]`]]
\begin{align*} x -2 \sin \left (y\right )+3+\left (2 x -4 \sin \left (y\right )-3\right ) \cos \left (y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.045 (sec). Leaf size: 22
ode:=x-2*sin(y(x))+3+(2*x-4*sin(y(x))-3)*cos(y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \arcsin \left (\frac {9 \operatorname {LambertW}\left (\frac {c_1 \,{\mathrm e}^{-\frac {1}{3}-\frac {8 x}{9}}}{9}\right )}{8}+\frac {3}{8}+\frac {x}{2}\right )
\]
✓ Mathematica. Time used: 60.476 (sec). Leaf size: 73
ode=(x-2*Sin[y[x]]+3)+(2*x-4*Sin[y[x]]-3)*Cos[y[x]]*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \arcsin \left (\frac {1}{8} \left (9 W\left (-\frac {1}{9} e^{-\frac {2}{9} (4 x+3-8 c_1)}\right )+4 x+3\right )\right ) \\
y(x)\to \arcsin \left (\frac {1}{8} \left (9 W\left (-\frac {1}{9} e^{-\frac {2}{9} (4 x+3-8 c_1)}\right )+4 x+3\right )\right ) \\
\end{align*}
✓ Sympy. Time used: 130.700 (sec). Leaf size: 364
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x + (2*x - 4*sin(y(x)) - 3)*cos(y(x))*Derivative(y(x), x) - 2*sin(y(x)) + 3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}}}{9 e^{\frac {1}{3}}}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}} e^{- \frac {1}{3} - \frac {2 i \pi }{9}}}{9}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (- \frac {\sqrt [9]{C_{1} e^{- 8 x}} e^{- \frac {1}{3} - \frac {i \pi }{9}}}{9}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (- \frac {\sqrt [9]{C_{1} e^{- 8 x}} e^{- \frac {1}{3} + \frac {i \pi }{9}}}{9}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}} e^{- \frac {1}{3} + \frac {2 i \pi }{9}}}{9}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}} \left (-1 - \sqrt {3} i\right )}{18 e^{\frac {1}{3}}}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}} \left (-1 + \sqrt {3} i\right )}{18 e^{\frac {1}{3}}}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}} \left (\sin {\left (\frac {\pi }{18} \right )} - i \cos {\left (\frac {\pi }{18} \right )}\right )}{9 e^{\frac {1}{3}}}\right )}{8} + \frac {3}{8} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {x}{2} + \frac {9 W\left (\frac {\sqrt [9]{C_{1} e^{- 8 x}} \left (\sin {\left (\frac {\pi }{18} \right )} + i \cos {\left (\frac {\pi }{18} \right )}\right )}{9 e^{\frac {1}{3}}}\right )}{8} + \frac {3}{8} \right )}\right ]
\]