plz solve all parts for this differential equations qs 3 Con

plz solve all parts for this differential equations qs

3. Consider the IVP: y -ty yo 1. a. Write down the equations for Euler\'s Method using h to denote the step size b. Verify that y e 2 is a solution of the given IVP

Solution

y\' = -ty , y(0) = 1

dy/dt = -ty

dy/y = -tdt

integrate both sides

=> ln(y) = -t^2/2 + C

y = e^[-t^2/2 + C]

y = e^(-t^2/2)*e^C

y = D*e^[-1/2*t^2]

use y(0) = 1

=> 1 = D*e^(0) , => D = 1

Hence y = e^[-1/2*t^2]    , -----------> solution to part (b)

In order to use Euler’s Method we first need to rewrite the differential equation into the form

y\' = -ty

we can see that f(t,y) = -ty

and to = 1 , yo = 1

now fo(0,1) = 0

and y1 = yo + hfo = 1 + h*0 = 1