Question

for differential equations

1. Identify each of the following differential equations as either Separable, Homogeneous, Linear Bernoulli, or Exact and solve the equation using the method of the type you have identified. Many can be classified in multiple ways, it is not necessary to list all possibilities.

Answer 1

implies (3x y^2 + 2y cos x) dy + (Y^3 - y^2 sin x - x) dx = 0 this is of the poem m dy + N dy = 0 where M = y^3 - y^2 sin - x and N = 3x y^2 + 2y cos x partial differential M/ partial differential y = 3y^2 - 2y sin x, partial differential N/ partial differential x = 3y^2 - 2y sin x partial differential M/ partial differential y = partial differential N/ partial differential x Therefore type exact differential equation solution is integrals M dx + integrals N dy = c implies integrals (y^3 - y^2 sin x - x) dx + integrals 0 dy = c implies x y^3 + y^2 cos x - x^2/2 = c is the general solution (4 x y + x^2 y) dy/dx = 2x + x y^2 implies y (4x + x^2) dy/dx = x (2 + y^2) implies y/2 + y^2 dy = x/4x + x^2 dx type: separable integrals y/2 + y^2 dy = integrals x/x (4 + x) dx implies 1/2 ln (2 + y^2) = ln (4 + x) + ln c_1 implies ln (2 + y^2) = 2 ln c_1 (4 + x) implies (2 + y^2) = c (4 + x)^2