Mister Exam

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

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

from to

### The solution

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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
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