Final answer:
To solve the given differential equation (y^3 + 6xy^4)dx + (3xy^2 + 12x^2y^3)dy = 0, we use the technique of exact differential equations. First, we check for exactness, then find the integrating factor. Multiplying the equation by the integrating factor, we simplify and integrate to obtain the solution e^6x*y^7 = C.
Step-by-step explanation:
To solve the given differential equation:
(y^3 + 6xy^4)dx + (3xy^2 + 12x^2y^3)dy = 0
We will use the technique of exact differential equations. First, we check if the equation is exact by calculating the partial derivatives of both terms:
∂(y^3 + 6xy^4)/∂y = 3y^2 + 24xy^3
∂(3xy^2 + 12x^2y^3)/∂x = 3y^2 + 12xy^3
Since the partial derivatives are equal, the equation is exact. Next, we find the integrating factor:
μ(x,y) = e^∫(∂M/∂y - ∂N/∂x) dx = e^∫(24xy^3 - 12xy^3) dx = e^∫12xy^3 dx = e^6x*y^4
Multiplying the entire equation by the integrating factor, we get:
e^6x*y^4(y^3 + 6xy^4)dx + e^6x*y^4(3xy^2 + 12x^2y^3)dy = 0
Simplifying the equation and rearranging terms, we obtain:
d(e^6x*y^7)/(dx) = 0
Now, we integrate both sides:
e^6x*y^7 = C
Where C is the constant of integration.