2*x^2*y*y'+y^2=2

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2 2 d y (x) + 2*x *--(y(x))*y(x) = 2 dx

$$2 x^{2} y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + y^{2}{\left(x \right)} = 2$$

2*x^2*y*y' + y^2 = 2

Detail solution

Given the equation:

$$2 x^{2} y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + y^{2}{\left(x \right)} = 2$$

This differential equation has the form:

where

$$f_{1}{\left(x \right)} = 1$$

$$g_{1}{\left(y \right)} = 1$$

$$f_{2}{\left(x \right)} = - \frac{1}{x^{2}}$$

$$g_{2}{\left(y \right)} = - \frac{2 - y^{2}{\left(x \right)}}{2 y{\left(x \right)}}$$

We give the equation to the form:

Divide both parts of the equation by g_2(y)

$$- \frac{2 - y^{2}{\left(x \right)}}{2 y{\left(x \right)}}$$

we get

$$\frac{2 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{1}{x^{2}}$$

We separated the variables x and y.

Now, multiply the both equation sides by dx,

then the equation will be the

$$\frac{2 dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{dx}{x^{2}}$$

or

$$\frac{2 dy y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{dx}{x^{2}}$$

Take the integrals from the both equation sides:

- the integral of the left side by y,

- the integral of the right side by x.

$$\int \frac{2 y}{y^{2} - 2}\, dy = \int \left(- \frac{1}{x^{2}}\right)\, dx$$

Detailed solution of the integral with y

Detailed solution of the integral with x

Take this integrals

$$\log{\left(y^{2} - 2 \right)} = Const + \frac{1}{x}$$

Detailed solution of the equation

We get the simple equation with the unknown variable y.

(Const - it is a constant)

Solution is:

$$y_{1} = y{\left(x \right)} = - \sqrt{C_{1} e^{\frac{1}{x}} + 2}$$

$$y_{2} = y{\left(x \right)} = \sqrt{C_{1} e^{\frac{1}{x}} + 2}$$

$$2 x^{2} y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + y^{2}{\left(x \right)} = 2$$

This differential equation has the form:

f1(x)*g1(y)*y' = f2(x)*g2(y),

where

$$f_{1}{\left(x \right)} = 1$$

$$g_{1}{\left(y \right)} = 1$$

$$f_{2}{\left(x \right)} = - \frac{1}{x^{2}}$$

$$g_{2}{\left(y \right)} = - \frac{2 - y^{2}{\left(x \right)}}{2 y{\left(x \right)}}$$

We give the equation to the form:

g1(y)/g2(y)*y'= f2(x)/f1(x).

Divide both parts of the equation by g_2(y)

$$- \frac{2 - y^{2}{\left(x \right)}}{2 y{\left(x \right)}}$$

we get

$$\frac{2 y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{1}{x^{2}}$$

We separated the variables x and y.

Now, multiply the both equation sides by dx,

then the equation will be the

$$\frac{2 dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{dx}{x^{2}}$$

or

$$\frac{2 dy y{\left(x \right)}}{y^{2}{\left(x \right)} - 2} = - \frac{dx}{x^{2}}$$

Take the integrals from the both equation sides:

- the integral of the left side by y,

- the integral of the right side by x.

$$\int \frac{2 y}{y^{2} - 2}\, dy = \int \left(- \frac{1}{x^{2}}\right)\, dx$$

Detailed solution of the integral with y

Detailed solution of the integral with x

Take this integrals

$$\log{\left(y^{2} - 2 \right)} = Const + \frac{1}{x}$$

Detailed solution of the equation

We get the simple equation with the unknown variable y.

(Const - it is a constant)

Solution is:

$$y_{1} = y{\left(x \right)} = - \sqrt{C_{1} e^{\frac{1}{x}} + 2}$$

$$y_{2} = y{\left(x \right)} = \sqrt{C_{1} e^{\frac{1}{x}} + 2}$$

The answer
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___________ / 1 / - / x y(x) = -\/ 2 + C1*e

$$y{\left(x \right)} = - \sqrt{C_{1} e^{\frac{1}{x}} + 2}$$

___________ / 1 / - / x y(x) = \/ 2 + C1*e

$$y{\left(x \right)} = \sqrt{C_{1} e^{\frac{1}{x}} + 2}$$

Graph of the Cauchy problem

The classification

factorable

separable

1st exact

Bernoulli

almost linear

lie group

separable Integral

1st exact Integral

Bernoulli Integral

almost linear Integral

Numerical answer
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(x, y):

(-10.0, 0.75)

(-7.777777777777778, 0.7765245094386053)

(-5.555555555555555, 0.8203782204706953)

(-3.333333333333333, 0.9072348027785913)

(-1.1111111111111107, 1.1636536929524415)

(1.1111111111111107, 1.4142135625688081)

(3.333333333333334, 1.4142135625316252)

(5.555555555555557, 1.4142135625172005)

(7.777777777777779, 1.4142135625168282)

(10.0, 1.414213562516456)

(10.0, 1.414213562516456)