Substitution methods and exact equations

Question

Assignment:

Substitution Methods and Exact Equations

Homogeneous Equations:

Dy/dx = F(y/x)

v = y/x,  y = vx,  dy/dx = v + x(dv/dx)

x(dv/dx) = F(v) - v

Bernoulli Equations:

dv/dx + (1-n) P(x)v = (1-n) Q(x)

Criterion for Exactness:

F(x,y) = ∫ M(x, y) dx + g(y)

Verify that the given differential equation is exact; then solve it.

Q1. (2x + 3y) dx + (3x + 2y) dy = 0

Q2. (x³ + y/x) dx + (y² + lnx) dy = 0

Q3. (x + tan^-1y) dx + ((x +y) / (1 + y²)) dy = 0

Provide complete and step by step solution for the question and show calculations and use formulas.