### Other calculators # Differential equation dy/dx=sin(y)

y() =
y'() =
y''() =
y'''() =
y''''() =

from to

### The solution

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d
--(y(x)) = sin(y(x))
dx                  
$$\frac{d}{d x} y{\left(x \right)} = \sin{\left(y{\left(x \right)} \right)}$$
y' = sin(y)
Detail solution
Given the equation:
$$\frac{d}{d x} y{\left(x \right)} = \sin{\left(y{\left(x \right)} \right)}$$
This differential equation has the form:
$$f_1(x)\ g_1(y)\ y' = f_2(x)\ g_2(y)$$
where
$$f_{1}{\left(x \right)} = 1$$
$$g_{1}{\left(y \right)} = 1$$
$$f_{2}{\left(x \right)} = 1$$
$$g_{2}{\left(y \right)} = \sin{\left(y{\left(x \right)} \right)}$$
We give the equation to the form:
$$\frac{g_1(y)}{g_2(y)}\ y'= \frac{f_2(x)}{f_1(x)}$$
Divide both parts of the equation by $g_{2}{\left(y{\left(x \right)} \right)}$
$$\sin{\left(y{\left(x \right)} \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{\sin{\left(y{\left(x \right)} \right)}} = 1$$
We separated the variables x and y.

Now, multiply the both equation sides by dx,
then the equation will be the
$$\frac{dx \frac{d}{d x} y{\left(x \right)}}{\sin{\left(y{\left(x \right)} \right)}} = dx$$
or
$$\frac{dy}{\sin{\left(y{\left(x \right)} \right)}} = dx$$

Take the integrals from the both equation sides:
- the integral of the left side by y,
- the integral of the right side by x.
$$\int \frac{1}{\sin{\left(y \right)}}\, dy = \int 1\, dx$$
Detailed solution of the integral with y
Detailed solution of the integral with x
Take this integrals
$$\frac{\log{\left(\cos{\left(y \right)} - 1 \right)}}{2} - \frac{\log{\left(\cos{\left(y \right)} + 1 \right)}}{2} = Const + x$$
Detailed solution of the equation
We get the simple equation with the unknown variable y.
(Const - it is a constant)

Solution is:
$$y_{1} = y{\left(x \right)} = - \operatorname{asin}{\left(\frac{1}{\tanh{\left(C_{1} + x \right)}} \right)} + \frac{3 \pi}{2}$$
$$y_{2} = y{\left(x \right)} = \operatorname{acos}{\left(- \frac{1}{\tanh{\left(C_{1} + x \right)}} \right)}$$
$$y\left(x\right)={\it ilt}\left({{\mathcal{L}\left(\sin y\left(x \right) , x , g_{19164}\right)+y\left(0\right)}\over{g_{19164}}} , g_{19164} , x\right)$$
y = 'ilt(('laplace(sin(y),x,g19164)+y(0))/g19164,g19164,x)
             /     1      \   3*pi
y(x) = - asin|------------| + ----
\tanh(C1 + x)/    2  
$$y{\left(x \right)} = - \operatorname{asin}{\left(\frac{1}{\tanh{\left(C_{1} + x \right)}} \right)} + \frac{3 \pi}{2}$$
           /    -1      \
y(x) = acos|------------|
\tanh(C1 + x)/
$$y{\left(x \right)} = \operatorname{acos}{\left(- \frac{1}{\tanh{\left(C_{1} + x \right)}} \right)}$$
Graph of the Cauchy problem
The classification
1st exact
1st exact Integral
1st power series
lie group
separable
separable Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 2.60429004329565)
(-5.555555555555555, 3.081941471758727)
(-3.333333333333333, 3.135126475764566)
(-1.1111111111111107, 3.140891927266994)
(1.1111111111111107, 3.141516714682852)
(3.333333333333334, 3.1415844258764265)
(5.555555555555557, 3.141591759127831)
(7.777777777777779, 3.141592558058867)
(10.0, 3.1415926427166294)
(10.0, 3.1415926427166294)
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