Question

X*sqrt(1-y^2)dx+sqrt(1-x^2)dy=0

Answer 1

Ответ:

$x\cdot \sqrt{1-y^2}\, dx+\sqrt{1-x^2}\, dy=0\\\\\\x\cdot \sqrt{1-y^2}\, dx=-\sqrt{1-x^2}\, dy\ \ ,\ \ \ \ \displaystyle \int\frac{x\, dx}{\sqrt{1-x^2}}=-\int \frac{dy}{\sqrt{1-y^2}}\\\\\\\int \frac{dy}{\sqrt{1-y^2}}=\frac{1}{2}\int \frac{-2x\, dx}{\sqrt{1-x^2}}\\\\\\arcsin\, y=\frac{1}{2}\cdot 2\sqrt{1-x^2}+C\ \ ,\ \ \ \ arcsin\, y=\sqrt{1-x^2}+C\ \ ,\\\\\\y=sin(\sqrt{1-x^2}+C)$

Answer 2

The given equation is a differential equation of the form Mdx + Ndy = 0. This is a first order linear differential equation, which can be solved using the method of integrating factors.

The equation is:

X*sqrt(1-y^2)dx + sqrt(1-x^2)dy = 0

First, we need to separate the variables. We can do this by dividing each term by its respective variable:

X*sqrt(1-y^2)/X = -sqrt(1-x^2)/y

This simplifies to:

sqrt(1-y^2) = -sqrt(1-x^2)/y

Now, we can solve this equation step-by-step:

1. Square both sides to remove the square root:

1 - y^2 = (1 - x^2)/y^2

2. Multiply both sides by y^2 to isolate y^2 on one side:

y^2(1 - y^2) = 1 - x^2

3. Factor out y^2:

y^2 - y^4 = 1 - x^2

4. Rearrange the terms:

y^4 + y^2 - 1 = x^2

5. Now, we can solve this equation for y. One way to do this is by completing the square on the left side of the equation. However, this is a fourth degree polynomial, which can be quite complex to solve.

Unfortunately, the search results provided do not contain a direct solution to this equation, so it might be necessary to use numerical methods or software to solve it.

Remember that the solutions to a differential equation can be complex, depending on the original equation. It's also important to note that the solutions may not be unique, as there could be multiple values of y that satisfy the equation for a given value of x.

Please consult a mathematics textbook or use a mathematical software tool for a more detailed and accurate solution.