How to use Monte Carlo Method

Please describe how to use the Monte Carlo method to estimate the attached expression. Thanks

Euler Method solved analytically

I have a differential equation with the initial condition given by: dy/dx=y^2/x+1, where y(0)= 1. (see attached file for more detail). As requested by my question, I have used the simple and improved euler methods to estimate y(1.2) with a step size of h=0.3 to 4 decimal places. I am struggling to solve the differential equation analytically. Can you help? I know that y(1.2) is 4.7272.

Numerical Representation & Errors - Write Effective Algorithm

See attachments for case study and complete questions. ...there exists a double series representation {see attachment}. This double series converges slowly, and it contains many small terms whose joint effect may be big. Your task is to write an effective algorithm to calculate the values of w(p,s) for any given {see attachment} Thanks for your help!

Polynomial - 3 Part Question

1 (a) Consider the function values... Consider a polynomial P(x) of least degree (the osculating polynomial) through the points xi=x0+i*h, i.e. polynomial that satisfies P(x0)=f0, P(x1)=f1, P'(x1)=f'(x1), P(x2)=f2 (b) Prove that df(x1)/dx = dP(x1)/dx for any smooth function f(x) (c) Construct the polynomial P(x) for the function f(x)=sin(x) and x0=0, x1=pi/2, x2=pi Please see attached for Full Question.

Numerical methods - Integration rule

Derive an integration rule for the domain [0,1] based on the quadrature points x1=0, x2=1/3 and x3=1, which is exact for polynomials of degree <= 2. Please see attached for full question.