Question

Solve the following differential equation:(y-x²)dx +(x+y²)dy=0 ________constant.

Answer 1

The given differential equation (y - x²)dx + (x + y²)dy = 0 is solved by verifying that it is an exact equation, then integrating and combining the results to find the general solution yx - (1/3)x³ + (1/3)y³ = C, where C is a constant.

Step-by-step explanation:

The differential equation given is:

(y - x²)dx + (x + y²)dy = 0

To solve this first-order differential equation, we'll attempt to see if it is an exact equation. An exact differential equation is in the form M(x,y)dx + N(x,y)dy = 0, where ∂M/∂y = ∂N/∂x.

In our given equation:

• M(x,y) = y - x²
• N(x,y) = x + y²

The partial derivatives are:

• ∂M/∂y = 1
• ∂N/∂x = 1

Since ∂M/∂y equals ∂N/∂x, the equation is exact.

Now, we integrate M with respect to x to find the function Φ(x,y):

∫ (y - x²) dx = yx - (1/3)x³ + f(y)

And we integrate N with respect to y to find the function Φ(x,y):

∫ (x + y²) dy = xy + (1/3)y³ + g(x)

From these two equations, we can determine that Φ(x,y) = yx - (1/3)x³ + (1/3)y³ = C, where C is a constant.

This is the general solution to the differential equation.