Find the general solution F(x,y)=C of the differential equation:
(-3(x^2)y-8x)dy+(-3xy^2-8y)dx=0
Where F(x,y)=___________________
To find the general solution of the given differential equation, we will use the method of exact differential equations.
Step 1: Check for Exactness
The given equation can be written in the form M(x, y)dx + N(x, y)dy = 0, where M(x, y) = -3(x^2)y - 8x and N(x, y) = -3xy^2 - 8y.
Now, compute the partial derivatives of M and N with respect to y (My) and x (Nx):
My = -3(x^2)
Nx = -3y^2
Step 2: Check for Exactness Condition
For a differential equation to be exact, it must satisfy the condition My = Nx. In this case, we see that My ≠ Nx.
Step 3: Introducing Integrating Factor
To make the equation exact, we need to introduce an integrating factor. The integrating factor, denoted by μ(x, y), can be found using the equation:
μ(x, y) = e^(∫[My - Nx] / Nx dx)
In this case, μ(x, y) = e^(∫[(-3(x^2)) - (-3y^2)] / (-3y^2) dx)
Simplifying the integral gives us: μ(x, y) = e^(∫[(y^2 - x^2) / y^2] dx)
Step 4: Multiplying the Equation by the Integrating Factor
Multiply the given equation by the integrating factor μ(x, y) to make it an exact equation:
(e^(∫[(y^2 - x^2) / y^2] dx))[(-3(x^2)y - 8x)dy + (-3xy^2 - 8y)dx] = 0
Simplifying this equation gives us: (-3(x^2)y - 8x)e^(∫[(y^2 - x^2) / y^2] dx)dy + (-3xy^2 - 8y)e^(∫[(y^2 - x^2) / y^2] dx)dx = 0
Step 5: Find Exact Differential
Now, we can observe that the equation is exact. Let's call the multiplier e^(∫[(y^2 - x^2) / y^2] dx) as μ(x, y).
μ(x, y) = e^(∫[(y^2 - x^2) / y^2] dx)
Now, we find the exact differential of μ(x, y)dx. Use the chain rule and multiply the partial derivative of μ(x, y) with respect to x (μx) by dx:
d(μ(x, y)dx) = μx(x, y)dx + μy(x, y)dy
Let's compute the partial derivatives of μ(x, y):
μx = [(y^2 - x^2) / y^2]
μy = 0 (as μ(x, y) is not explicitly dependent on y)
Therefore, d(μ(x, y)dx) = [(y^2 - x^2) / y^2]dx
Step 6: Rewrite the Equation as an Exact Differential
Now, rewrite the given equation using the exact differential:
(-3(x^2)y - 8x)e^(∫[(y^2 - x^2) / y^2] dx)dy + (-3xy^2 - 8y)e^(∫[(y^2 - x^2) / y^2] dx)dx = d(μ(x, y)dx)
Step 7: Integrate to Find the General Solution
Integrating both sides of the equation will give us the general solution:
∫[-3(x^2)y - 8x]e^(∫[(y^2 - x^2) / y^2] dx)dy + ∫[-3xy^2 - 8y]e^(∫[(y^2 - x^2) / y^2] dx)dx = ∫[(y^2 - x^2) / y^2]dx
Integrating each term separately will result in a general solution of the differential equation:
F(x, y) = ∫[-3(x^2)y - 8x]e^(∫[(y^2 - x^2) / y^2] dx)dy + ∫[-3xy^2 - 8y]e^(∫[(y^2 - x^2) / y^2] dx)dx + C
where C is the constant of integration.