Solving the bernoulli equation

Question

Assignment:

Q1. Find solutions to the given Cauchy- Euler equation

(a)  xy'+ y =0      (b)  x²y'' + xy'+y  =0 ; y(1)  =1,  y'(1)  =0

Q2. Find a solution to the initial value problem

x²y' + 2xy = 0;   y (1) = 2

Q3. Find the general solution to the given problems

(a)  Y' + (cot x)y  =  2cosx          (b)     (x-5)(xy'+3y) = 2

Q4. Solve the Bernoulli equation   y'  = xy³ - 4y

Q5. Use separation of variables to solve the verhulst population problem

N' (t) = (a-bN) N, N (0) =  N₀;   a,b > 0

Q6. Verify that each of the given functions is a solution of the given differential equation, and then use the Wronskian to determine linear dependence/ independence

Y''' - y''- 2y' = 0       {1, e^-x, e^2x}

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