Mister Exam

Differential equation y'+2sin(2pix)=0

For Cauchy problem:

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

The graph:

from to

The solution

You have entered [src]
                d           
2*sin(2*pi*x) + --(y(x)) = 0
                dx          
$$2 \sin{\left(2 \pi x \right)} + \frac{d}{d x} y{\left(x \right)} = 0$$
2*sin(2*pi*x) + y' = 0
Detail solution
Given the equation:
y' = $$- 2 \sin{\left(2 \pi x \right)}$$
This differential equation has the form:
y' = f(x)

It is solved by multiplying both sides of the equation by dx:
y'dx = f(x)dx, or

d(y) = f(x)dx

And by using the integrals of the both equation sides:
∫ d(y) = ∫ f(x) dx

or
y = ∫ f(x) dx

In this case,
f(x) = $$- 2 \sin{\left(2 \pi x \right)}$$
Consequently, the solution will be
y = $$\int \left(- 2 \sin{\left(2 \pi x \right)}\right)\, dx$$
Detailed solution of the integral
or
y = $$\frac{\cos{\left(2 \pi x \right)}}{\pi}$$ + C1
where C1 is constant, independent of x
The answer [src]
            cos(2*pi*x)
y(x) = C1 + -----------
                 pi    
$$y{\left(x \right)} = C_{1} + \frac{\cos{\left(2 \pi x \right)}}{\pi}$$
Graph of the Cauchy problem
The classification
nth algebraic
separable
1st exact
1st linear
1st power series
lie group
nth linear constant coeff undetermined coefficients
nth linear constant coeff variation of parameters
nth linear euler eq nonhomogeneous variation of parameters
nth algebraic Integral
separable Integral
1st exact Integral
1st linear Integral
nth linear constant coeff variation of parameters Integral
nth linear euler eq nonhomogeneous variation of parameters Integral
Numerical answer [src]
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.4869641616583301)
(-5.555555555555555, 0.13257687557567086)
(-3.333333333333333, 0.27253531157417477)
(-1.1111111111111107, 0.6755298423220262)
(1.1111111111111107, 0.6755298980576385)
(3.333333333333334, 0.27253538393302046)
(5.555555555555557, 0.13257680464730812)
(7.777777777777779, 0.4869642732536753)
(10.0, 0.750000196116789)
(10.0, 0.750000196116789)
The graph
Differential equation y'+2sin(2pix)=0
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