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

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

y*y' = -x

Detail solution

Given the equation:

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

This differential equation has the form:

$$f_1(x)*g_1(y)*y' = f_2(x)*g_2(y)$$

where

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

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

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

$$g_{2}{\left(y \right)} = \frac{1}{y{\left(x \right)}}$$

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 $g_{2}{\left(y{\left(x \right)} \right)}$

$$\frac{1}{y{\left(x \right)}}$$

we get

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

We separated the variables x and y.

Now, multiply the both equation sides by dx,

then the equation will be the

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

or

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

Take the integrals from the both equation sides:

- the integral of the left side by y,

- the integral of the right side by x.

$$\int y\, dy = \int \left(- x\right)\, dx$$

Detailed solution of the integral with y

Detailed solution of the integral with x

Take this integrals

$$\frac{y^{2}}{2} = - \frac{x^{2}}{2} + Const$$

Detailed solution of the equation

We get the simple equation with the unknown variable y.

(Const - it is a constant)

Solution is:

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

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

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

This differential equation has the form:

$$f_1(x)*g_1(y)*y' = f_2(x)*g_2(y)$$

where

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

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

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

$$g_{2}{\left(y \right)} = \frac{1}{y{\left(x \right)}}$$

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 $g_{2}{\left(y{\left(x \right)} \right)}$

$$\frac{1}{y{\left(x \right)}}$$

we get

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

We separated the variables x and y.

Now, multiply the both equation sides by dx,

then the equation will be the

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

or

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

Take the integrals from the both equation sides:

- the integral of the left side by y,

- the integral of the right side by x.

$$\int y\, dy = \int \left(- x\right)\, dx$$

Detailed solution of the integral with y

Detailed solution of the integral with x

Take this integrals

$$\frac{y^{2}}{2} = - \frac{x^{2}}{2} + Const$$

Detailed solution of the equation

We get the simple equation with the unknown variable y.

(Const - it is a constant)

Solution is:

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

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

The answer
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_________ / 2 y(x) = -\/ C1 - x

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

_________ / 2 y(x) = \/ C1 - x

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

Graph of the Cauchy problem

The classification

1st exact

1st exact Integral

1st homogeneous coeff best

1st homogeneous coeff subs dep div indep

1st homogeneous coeff subs dep div indep Integral

1st homogeneous coeff subs indep div dep

1st homogeneous coeff subs indep div dep Integral

1st power series

Bernoulli

Bernoulli Integral

factorable

lie group

separable

separable Integral

Numerical answer
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(x, y):

(-10.0, 0.75)

(-7.777777777777778, 6.329982240581089)

(-5.555555555555555, 8.348551178571897)

(-3.333333333333333, 9.457874778337375)

(-1.1111111111111107, 9.966340355961792)

(1.1111111111111107, 9.96634039631456)

(3.333333333333334, 9.457874568925822)

(5.555555555555557, 8.348550566706328)

(7.777777777777779, 6.329980785921501)

(10.0, 0.7499815578513588)

(10.0, 0.7499815578513588)

The graph