a$.$

How can we derive the Secant method from the Newton$-$Raphson method?

b$.$ Find the lower triangular matrix $L$ and the upper triangular $U$ in the $[L][U]$

decomposition of the matrix given below:

a$.$ What is the order of error of Runge$-$Kutta $2^{nd}$ Order Method for solving ODEs in relation to

step size $\text{'}$h$\text{'}?$ How can it be explained using Taylor series?

b$.$ For the ordinary differential equation

$2\frac{dy}{dx}+3y=e^{-1\mathrm{.}5x}$;$y(0)=5$

Use Runge$-$Kutta $2$nd Order $($Midpoint Method$)$ to find $y(2\mathrm{.}5)$ using $h=1\mathrm{.}25$

OR

a$.$ What is the order of error of Euler's Method for solving ODEs in relation to the step size $\text{'}h\text{'}$

How can it be explained using the Taylor series?

b$.$ Use Euler's method with $h=0\mathrm{.}1$ to find approximate values for the solution of the initial va

problem at $x=0\mathrm{.}2$

$\frac{dy}{dx}+2y=x*e^{-2x}$;$y(0)=1$