### Other calculators # Differential equation dx+xydy=(y^2)dx+ydy

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

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### The solution

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      d                2      d
1 + x*--(y(x))*y(x) = y (x) + --(y(x))*y(x)
dx                      dx           
$$x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 1 = y^{2}{\left(x \right)} + y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}$$
x*y*y' + 1 = y^2 + y*y'
Detail solution
Given the equation:
$$x y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} - y^{2}{\left(x \right)} - y{\left(x \right)} \frac{d}{d x} y{\left(x \right)} + 1 = 0$$
This differential equation has the form:
f1(x)*g1(y)*y' = f2(x)*g2(y),

where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{x - 1}$$
$$\operatorname{g_{2}}{\left(y \right)} = - \frac{y^{2}{\left(x \right)} - 1}{y{\left(x \right)}}$$
We give the equation to the form:
g1(y)/g2(y)*y'= f2(x)/f1(x).

Divide both parts of the equation by g2(y)
$$- \frac{y^{2}{\left(x \right)} - 1}{y{\left(x \right)}}$$
we get
$$- \frac{y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{1}{x - 1}$$
We separated the variables x and y.

Now, multiply the both equation sides by dx,
then the equation will be the
$$- \frac{dx y{\left(x \right)} \frac{d}{d x} y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{dx}{x - 1}$$
or
$$- \frac{dy y{\left(x \right)}}{y^{2}{\left(x \right)} - 1} = - \frac{dx}{x - 1}$$

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

Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = - \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
$$\operatorname{y_{2}} = y{\left(x \right)} = \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
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y(x) = -\/  1 + C1 + C1*x  - 2*C1*x 
$$y{\left(x \right)} = - \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
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y(x) = \/  1 + C1 + C1*x  - 2*C1*x 
$$y{\left(x \right)} = \sqrt{C_{1} x^{2} - 2 C_{1} x + C_{1} + 1}$$
Graph of the Cauchy problem
The classification
separable
1st exact
1st power series
lie group
separable Integral
1st exact Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.8493599517053494)
(-5.555555555555555, 0.919028857717191)
(-3.333333333333333, 0.965455921734598)
(-1.1111111111111107, 0.9919100594412996)
(1.1111111111111107, 0.9999777939747626)
(3.333333333333334, 0.990169095120081)
(5.555555555555557, 0.9619883863793672)
(7.777777777777779, 0.9137373526062414)
(10.0, 0.8419725920923582)
(10.0, 0.8419725920923582)
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