Question

Xdx-ydy=yx^2dy+xy^2dx

Answer 1

Дано уравнение:

xdx - ydy = yx^2dy + xy^2dx

Давайте разберемся с ним:

Сгруппируем члены, содержащие dx, и члены, содержащие dy:

xdx - ydx = yx^2dy + xy^2dx

(x - y)dx = yx^2dy + xy^2dx

Теперь вынесем общие множители dx и dy:

(x - y)dx - xy^2dx = yx^2dy

[(x - y) - xy^2]dx = yx^2dy

Теперь разделим обе стороны на [(x - y) - xy^2]:

dx / dx = (yx^2dy) / [(x - y) - xy^2]

1 = (yx^2dy) / [(x - y) - xy^2]

Теперь выразим dy:

dy = [(x - y) - xy^2] / (yx^2) dx

Таким образом, решением данного дифференциального уравнения является выражение:

dy = [(x - y) - xy^2] / (yx^2) dx

Answer 2

To solve the given equation, we can rearrange it and simplify:

xdx - ydy = yx^2dy + xy^2dx

Rearranging, we get:

xdx + yx^2dy = ydy + xy^2dx

Now, let's separate the variables by moving all terms involving 'x' to one side and all terms involving 'y' to the other side:

xdx - xy^2dx = ydy - yx^2dy

Factoring out 'dx' on the left side and 'dy' on the right side:

x(dx - y^2dx) = y(dy - x^2dy)

Now, we can divide both sides by (dx - y^2dx) and (dy - x^2dy) respectively:

x = y/(dx - y^2dx) * (dy - x^2dy)

Next, we can simplify the expression further:

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

Dividing both sides by x(dy - x^2dy), we have:

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

Now, we can cross-multiply:

y = dx(1 - y^2)

Expanding the right side:

y = dx - dx*y^2

Moving all terms involving 'y' to one side:

dx*y^2 + y - dx = 0

This is a quadratic equation in terms of 'y'. To solve for 'y', we can use the quadratic formula:

y = (-b ± √(b^2 - 4ac))/(2a)

Comparing the quadratic equation to the general form ax^2 + bx + c = 0, we have:

a = dx b = 1 c = -dx

Plugging these values into the quadratic formula:

y = (-(1) ± √((1)^2 - 4(dx)(-dx)))/(2(dx))

Simplifying:

y = (-1 ± √(1 + 4dx^2))/(2dx)

Therefore, the solution to the given equation is:

y = (-1 ± √(1 + 4dx^2))/(2dx)