Question

(1+ y^2)dx – Vx*dy = 0V-это корень​

Answer 1

уравнение с разделяющимися переменными:

$\frac{dx}{\sqrt{x} } =\frac{dy}{1+y^2} \\\\\int\frac{dx}{\sqrt{x} }=\int\frac{dy}{1+y^2}\\\\2\sqrt{x}=arctgy+C$

Answer 2

I'm not entirely sure what you mean by "V-это корень" in the context of the given expression. The expression (1+ y^2)dx – Vx*dy = 0 is a first-order ordinary differential equation (ODE) that relates the variables x and y. The equation is in the form of a first-order linear ODE.

To solve this differential equation, we can rearrange it as follows:

(1+ y^2)dx = Vx*dy

Now, we can integrate both sides of the equation to find the general solution. However, without a specific initial condition or boundary condition, the solution will be in the form of an implicit function.

Integrating both sides:

∫(1+ y^2)dx = ∫Vx*dy

The integral on the left side with respect to x will be:

x + y^2x = ∫Vxdy

And we can integrate the right side with respect to y:

x + y^2*x = V∫xdy

The integral of x with respect to y is simply xy:

x + y^2*x = Vxy + C

where C is the constant of integration.

So, the general solution to the given differential equation is:

x + y^2*x = Vxy + C

If you have a specific initial condition or boundary condition, you can use it to determine the value of the constant C and obtain a unique solution. Otherwise, the solution remains implicit.