Solve the following equation numerically using the forward Euler method du =t²-10 with the initial condition u(0)=50 dt As part of the solution process converge the solution by decreasing the step size You can determine the solution by taking the root-mean-square of the error with respect to the exact solution. You will need to calculate the solution to this differential equation. Then for each step size h, you can calculate the root-mean-square of the error. Once the root-mean-square of the error is less than the accuracy of the solution, your result is converged. Choose a time grid say from 0 to 20. Once you have this working, try solving using the Forward Euler method, du dt =u*cos(t) with the initial condition u(0)=1, and using a mesh from [0,10].

(Numerical analysis) Here’s a challenging problem for those who know a little calculus. The Newton-Raphson method can be used to find the roots of any equation y(x)=0. In this method, the (i+1)stapproximation,xi+1,toarootofy(x)=0 is given in terms of the ith approximation, xi, by the following formula, where y’ denotes the derivative of y(x) with respect to x: xi+1=xiy(xi)/y(xi) For example, if y(x)=3x2+2x2,theny(x)=6x+2 , and the roots are found by making a reasonable guess for a first approximation x1 and iterating by using this equation: xi+1=xi(3xi2+2xi2)/(6xi+2) a. Using the Newton-Raphson method, find the two roots of the equation 3x2+2x2=0. (Hint: There’s one positive root and one negative root.) b. Extend the program written for Exercise 6a so that it finds the roots of any function y(x)=0, when the function for y(x) and the derivative of y(x) are placed in the code.

Please no written by hand solutions Numerical Methods

using the R application

A 200 gallon tank initially contains 100 gallons of water with 20 pounds of salt. A salt solution with 1/5 pound of salt per gallon is added to the tank at 10 gal/min, and the resulting mixture is drained out at 5 gal/min. Let Q(t) denote the quantity (lbs) of salt at time t (min). (a) Write a differential equation for Q(t) which is valid up until the point at which the tank overflows. Q' (t) = = (b) Find the quantity of salt in the tank as it's about to overflow. esc C ✓ % 1 1 a 2 W S # 3 e d $ 4 f 5 rt 99 6 y & 7 h O u * 00 8 O 1 9 1 O

Write cpp which can solve equation using Eular method

of Find two solutions for the nonlinear equation below. Use the inline function and the suitable solver to solve it and find the two solutions. (Don't use scripts) rcos (1) −1

5

V Obtain the expression for y(t) which is satisfying the differential equation ÿ + 3y+ 2y = et y(0)=0 and y(0)=0

7. Solve with Python. A peristaltic pump delivers a unit flow (Q₁) of a highly viscous fluid. The network is depicted in the figure. Every pipe section has the same length and diameter. The mass and mechanical energy balance can be simplified to obtain the flows in every pipe. Solve the following system of equations to obtain the flow in every pipe using matrix inverse. S Q₂ 0₂ le 0₂ 90 Q₁+ 20-20-0 Qs+ 206-20-0 307-206-0