# Differential equation ylnydx+xdy=0

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

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### The solution

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

where
$$\operatorname{f_{1}}{\left(x \right)} = 1$$
$$\operatorname{g_{1}}{\left(y \right)} = 1$$
$$\operatorname{f_{2}}{\left(x \right)} = - \frac{1}{x}$$
$$\operatorname{g_{2}}{\left(y \right)} = y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}$$
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)
$$y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}$$
we get
$$\frac{\frac{d}{d x} y{\left(x \right)}}{y{\left(x \right)} \log{\left(y{\left(x \right)} \right)}} = - \frac{1}{x}$$
We separated the variables x and y.

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

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

Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = e^{\frac{C_{1}}{x}}$$
        C1
--
x
y(x) = e  
$$y{\left(x \right)} = e^{\frac{C_{1}}{x}}$$
Graph of the Cauchy problem
The classification
separable
1st exact
separable reduced
lie group
separable Integral
1st exact Integral
separable reduced Integral
(x, y):
(-10.0, 0.75)
(-7.777777777777778, 0.6908192813300311)
(-5.555555555555555, 0.5958133031715364)
(-3.333333333333333, 0.4218748991197924)
(-1.1111111111111107, 0.07508464091707924)
(1.1111111111111107, nan)
(3.333333333333334, nan)
(5.555555555555557, nan)
(7.777777777777779, nan)
(10.0, nan)
(10.0, nan)
To see a detailed solution - share to all your student friends
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share to all your student friends: