Question

Dx + (xy - y³)dy = 0​

Answer 1

Ответ снизу))))))))))))))

Answer 2

The given equation is a first-order ordinary differential equation. It is not in a standard form and does not appear to be separable or exact. However, we can try to solve it using an integrating factor.

To start, let's rearrange the equation:

dx + (xy - y³)dy = 0

To make it look more like a standard form, we can rewrite it as:

dx + xydy - y³dy = 0

Now, let's focus on the term involving y and try to find an integrating factor for it. The integrating factor is usually denoted as μ(x) and is defined as:

μ(x) = e^(∫P(x)dx)

where P(x) is the coefficient of dy. In this case, P(x) = x.

So, let's calculate the integrating factor:

μ(x) = e^(∫x dx) = e^(x²/2) [applying the integral]

Now, we multiply the entire equation by the integrating factor μ(x):

e^(x²/2)dx + xye^(x²/2)dy - y³e^(x²/2)dy = 0

Now, we need to check if the left-hand side of the equation can be written as the total derivative of some function with respect to y. Let's inspect the terms:

d(e^(x²/2)y) = e^(x²/2)dy + xe^(x²/2)dy

The left-hand side of our equation has similar terms, but with an additional -y³e^(x²/2)dy term. So, we need to find a function that has y³e^(x²/2)dy as its total derivative with respect to y.

Let's integrate y³e^(x²/2)dy with respect to y:

∫y³e^(x²/2)dy = (1/4)e^(x²/2)y⁴ + C(x)

where C(x) is the constant of integration with respect to y. We can now rewrite our equation using this result:

d(e^(x²/2)y) - (1/4)e^(x²/2)y⁴dy + C(x) = 0

Now, this equation can be written as a total derivative with respect to y:

d(e^(x²/2)y - (1/4)e^(x²/2)y⁴ + C(x)y) = 0

At this point, we can consider the equation to be equal to a constant, say K:

d(e^(x²/2)y - (1/4)e^(x²/2)y⁴ + C(x)y) = K

This is the general solution to the given differential equation. It involves an arbitrary constant C(x) that is determined by any initial or boundary conditions you may have.