differential equations

Question

Consider the differential equation x^4 y^4 dx + x^5 y^3 dy = 0

(a) Show that the equation is not exact.

(b) Show that µ = xy2 is an integrating factor.

(c) Find an implicit solution for the differential equation using the integrating factor.

Answer 1

a) x⁴y⁴dx + x⁵y³dy = 0;∂

∂(x⁴y⁴)/∂y = 4x⁴y³; ∂(x⁵y³)/∂x = 5x⁴y³.

Because ∂(x⁴y⁴)/∂y ≠ ∂(x⁵y³)/∂x this equation is not exact.

b) Now let multiply our equation by μ = xy². We get equation: x⁵y⁶dx + x⁶y⁵dy = 0;

∂(x⁵y⁶)/∂y = ∂(x⁶y⁵)/∂x = 6x⁵y⁵; So new equation is exact.

c)There exist function F(x, y) such that dF(x,y) = x⁵y⁶dx + x⁶y⁵dy = 0.∂

∂F/∂x = x⁵y⁶, F = ∫x⁵y⁶dx = x⁶y⁶/6 + g(y); then ∂F/∂y = x⁶y⁵ + g'(y) = x⁶y⁵; g'(y) = 0 and g(y) = C.

Such a way F(x,y) = x⁶y⁶/6 + C is implicit general solution