Determine the cauchy-euler equation

Question

Solve the following problems:

1. Find solutions to the given Cauchy- Euler equation

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

2. Find a solution to the initial value problem

x²y' + 2xy = 0; y (1) = 2

3. Find the general solution to the given problems

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

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

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

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

6. 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}