Solve the following problem for one step: dy dx = y + x² y(0)=1, 0≤x≤2, h=2 a) Use Euler's method to find y(2). b) Use Heun's method to find y(2).

Find a closed form solution for \max_{\theta \in C_{t-1}} <x,\theta>. You may want to use the Lagrange multiplier method.

I need the algorithm, proof of correctness and runtime analysis for the problem. No code necessary ONLY algorithm. And runtime should be O(n log n) as stated in the question.

(b) Let n be an integer. Show that n if the sum of the digits of n is a multiple of 3,then n is a multiple of 3. You must do so by a direct proof. For this problem youmay assume that ∀a, b ∈ Z, ∀m ∈ Z+, (a ≡ x (mod m) ∧ b ≡ y (mod m)) ⇒ ab ≡ xymod m. *Solve by direct proof*

Solve it , only if you are really very very good in it. Otherwise i will downvote. Computer science

Write computer codes using any programming language for the two-equation Newton-Raphson method and test it by solving problem. Use multiple equation Newton-Raphson method to determine one of the roots of the following simultaneous nonlinear equations within ɛ, = 0.01%. x² + y? = 16 (x – 4)² + (y – 4)² = 5

What is the value of x after the 5th iteration of performing fixed-point iteration on the equation f(x) = x3 –x – 2? (Use the equation where the root converges and use an initial guess of 1)   Group of answer choices 1.2599 1.5206 1.5158 1.5213

3. Simplify the following expression using Quine-McCluskey method?F(V,W, X,Y,Z) = ∑m(0,3,5,6,7,10,12,13,19,23, 27, 29) + ∑d(2,8,9,15,17,18,25)

Write computer codes using any programming language for one of the following methods to solve a linear equation system.1- Gaussian elimination.2- LU decomposition.3- Gauss-Seidel iterations.Execute the program for the linear equation system given below.