Assignment:
Q1. Suppose that n does not equal to zero and n does not equal to one. Show that the substitution v = y1-n transforms the Bernoulli equation dy/dx + P(x)y = Q(x)yn into the linear equation dv/dx + (1-n)P(x)v(x) = (1-n)Q(x).
Q2. The equation dy/dx = A(x)y2 + B(x)y + C(x) is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known.
Show that the substitution y = y1 + (1/v) transforms the Riccati equation into the linear equation dv/dx + (B +2Ay1)v = -A
Q3. An equation of the form y = xy' + g(y') is called a Clairaut equation. Show that the one parameter family of straight lines described by y(x) = Cx + g(C) is a general solution of y = xy' + g(y').
Provide complete and step by step solution for the question and show calculations and use formulas.