- [20 Pts] Calculate an approximate value for
**x(0.5)**where the function**x(t)**satisfies the ordinary differential equation- 2 3 | x' = t + x | x(0) = 0 -

- (a) using
**one step**of the Taylor Series method of order 2, - (b) using
**one step**of the Runge-Kutta method of order 2 (i.e., RK2).

- (a) using
- [20 Pts] Complete the following Matlab function
`euler.m`which could be used to solve the following initial-value problem (IVP) for**x(b)**.- | x' = f(t,x(t)) | x(a) = s -

using**n**steps of size**h=(b-a)/n**. You may assume the existence of an external Matlab function`func.m`which is invoked via func(t,x) to evaluate the function**f(t,x(t))**in the IVP.function [e] = euler(n,a,b,s) h = (b-a)/n;

## Turn page over ...

- [20 Pts] Write an equivalent
**autonomous**system of first-order differential equations for the following**coupled**system of first-order ordinary differential equations.- 3 y | x' = x + e + t | 2 | y' = log x - cos y + t x | | x(0) = 1, y(0) = 2 -

- [20 Pts] Rewrite the following third-order ordinary differential
equation as an autonomous system of first-order ordinary
differential equations having the form
**X'=F(X), X(1)=S**, where**X, X', F,**and**S**are all vectors.- | x''' = 2 x'' + cos(tx) - x'/t | x(1) = 1, x'(1) = 0, x''(1) = 2 -

- [20 Pts] Sketch the approximations generated by the
Euler and RK2 methods to the solution
curve
**x(t)**shown on the**last page**of the answer sheets for**t = a+ih**, where**a=1.0**,**h=1.0**, and**i=0,1,2**. That is, you must illustrate 3 approximate values for the the solution**x(t)**by each method.**Label each approximation clearly.**