Assignment:
Q1. Show that the substitution v = lny transforms the differential equation
dy/dx + P(x)y = Q(x)(ylny) into the linear equation dv/dx + P(x) = Q(x)v(x)
Q2. Use the idea in Problem 57 to solve the equation
x (dy/dx) - 4x2y + 2ylny = 0
Q3. Solve the differential equation dy/dx = (x-y-1)/(x+y+3) by finding h and k so that the substitutions x = u + h, y = v + k transform it into the homogeneous equation
dv/du = (u - v)/ (u + v)
Q4. Use the method in question 3 to solve the differential equation
dy/dx = (2y - x + 7) / (4x - 3y - 18)
Provide complete and step by step solution for the question and show calculations and use formulas.