For the initial value problem:

$$

$2$

$$

$2+4=0$ ; $(0)=0\mathrm{.}5$ and

$$

$$

$(0)=0$

a$)$ Write out the analytical solution:

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

b$)$ Using the analytical solution, calculate the value of

y

at

t $=0\mathrm{.}3$

: $\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

c$)$ Derive the Explicit Euler numerical solution algorithm for this problem. Write the recursion

formulae:

$+1=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_+1=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

d$)$ Using the Explicit Euler, calculate the value of

y

at

t $=0\mathrm{.}3$

using

$$t $=0\mathrm{.}1$

: $\_\_\_\_\_\_\_\_\_\_\_\_$

e$)$ Derive the Implicit Euler numerical solution algorithm for this problem. Write the recursion

formulae:

$+1=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_+1=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

f$)$ Using the Implicit Euler, calculate the value of

y

at

t $=0\mathrm{.}3$

using

$$t $=0\mathrm{.}1$

: $\_$