Question

Use one of these equations to solve!

1. Separation of variables:
2.Homogeneous equation
3. Exact equation

I need answer ASAP please!

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Answer 1

the solution to the differential equation is:

3y^2 - 12y = x^2 + 6.

Explanation:

We can use the equation 6y dy - 12 dy = 2xy dx to solve this differential equation using separation of variables.

First, we can rearrange the equation as:

(6y - 12)dy = 2xy dx

Next, we can separate the variables:

(6y - 12)dy = 2xy dx

∫ (6y - 12)dy = ∫ 2xy dx

3y^2 - 12y = x^2 + C

where C is the constant of integration.

Now we can use the initial condition to solve for C. Let's say the initial condition is y(0) = 2, then we have:

3(2)^2 - 12(2) = 0 + C

C = 6

Therefore, the solution to the differential equation is:

3y^2 - 12y = x^2 + 6.

This differential equation is not homogeneous or exact.