# Differential equation xydx-(x+2)dy=0

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

from to

### The solution

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

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

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

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

Solution is:
$$\operatorname{y_{1}} = y{\left(x \right)} = \frac{C_{1} e^{x}}{x^{2} + 4 x + 4}$$
              x
C1*e
y(x) = ------------
2
4 + x  + 4*x
$$y{\left(x \right)} = \frac{C_{1} e^{x}}{x^{2} + 4 x + 4}$$
The classification
factorable
separable
1st exact
1st linear
Bernoulli
almost linear
1st power series
lie group
separable Integral
1st exact Integral
1st linear Integral
Bernoulli Integral
almost linear Integral
To see a detailed solution - share to all your student friends
To see a detailed solution,
share to all your student friends: