How to find the exact solution from this equations?

1. (x+epsilon*y)dy/dx + y =0      y(1)=1

2. (x^n + epsilon * y)dy/dx + nx^(n-1) * y = m*x^(m-1)      y(1)=b>1   ; n=2,3,4,...   ; m=0,1,2,3,...

3. u" + u + epsilon*u^3 = 0    u(0)=A  ;   u'(0)=0

4. y" + 2(y')^2 +3y' = -sin x    ; y(0)=1  ;  y'(0)=0

non-linear and non homogen equation second-order..

1.) y"+y^2 = x^4 + 2

with 0<x<1, y(0)=0, y(1)=1

2.) y"+y^2 = -b*y^2 * y'

with y(0)=1, y'(0)=0 and b=0.1 ; 0.3 ; 0.5 ; 0.7

how to solve thats??

thanks in advance.

How do I know how many Eulerian circuits in this graph and what trajectory?

Thanks..

help me please for create this graph..

I've tried it, but that appears is a box shape

1. nonlinear ODE with parameter "epsilon"

(x^n +epsilon*y(x))dy/dx + n*x^(n-1) * y(x) =m*x^(m-1) ; y(1)=b>1

where n=2,3,4,.. and m=0,1,2,3,...

2. Duffing equation with parameter "epsilon"

d^2 y(x)/dx^2 + y(x) + epsilon*y(x)^3=0 ; y(0)=A ; y'(0)=0