Problem:
Solve the given differential equation: (y - x^2)dx + (y - x^2)dy = 0.

Solution:

To solve the given differential equation, we will use the method of exact differential equations. A differential equation is said to be exact if it can be written in the form M(x, y)dx + N(x, y)dy = 0, where M and N are functions of both x and y, and their partial derivatives with respect to y and x, respectively, are equal: ∂M/∂y = ∂N/∂x.

Step 1: Identifying M and N

In the given differential equation, (y - x^2)dx + (y - x^2)dy = 0, we can identify M = y - x^2 and N = y - x^2.

Step 2: Checking for Exactness

To check if the equation is exact, we need to verify if ∂M/∂y = ∂N/∂x.

∂M/∂y = 1
∂N/∂x = -2x

Since ∂M/∂y ≠ ∂N/∂x, the equation is not exact.

Step 3: Introducing the Integrating Factor

To make the equation exact, we need to find an integrating factor, denoted by μ(x, y), which is a function that multiplies both M and N to make the equation exact. The integrating factor is given by the formula:

μ(x, y) = e^(∫(∂M/∂x - ∂N/∂y)dx)

In this case, μ(x, y) = e^(∫(-2x - 1)dx)

Step 4: Finding the Integrating Factor

To find the integrating factor, we need to compute the integral in the formula for μ(x, y).

∫(-2x - 1)dx = -x^2 - x + C

Therefore, μ(x, y) = e^(-x^2 - x + C) = e^(-x^2 - x)e^C = Ce^(-x^2 - x)

Step 5: Multiplying the Equation by the Integrating Factor

Now, we multiply both sides of the given differential equation by the integrating factor μ(x, y):

Ce^(-x^2 - x)(y - x^2)dx + Ce^(-x^2 - x)(y - x^2)dy = 0

Simplifying this equation gives us:

Cy(e^(-x^2 - x)dx + e^(-x^2 - x)dy) - Cx^2(e^(-x^2 - x)dx + e^(-x^2 - x)dy) = 0

Step 6: Simplifying and Integrating

Now, we can see that the equation is exact, and we can write it as:

d(F(x, y)) = 0

where F(x, y) = Cy - Cx^2.

Integrating both sides, we get:

F(x,