How can I change this explicit method code to the implicit method code?

f = @(t,y) -100000*y + 99999*exp(-t);

t0 = 0; y0 =0;

h = 0.01; tn = 2;

n = (tn-t0)/h;

t(1)=t0; y(1)=y0;

for i=1:n

t(i+1) = t(i) + h;

y(i+1) = y(i) + (f(t(i),y(i))*h);

end

plot(t,y)

Answer 1

[David Goodmanson]

Hi MC

I am not sure what is meant by 'implicit method', and while numerical methods are good, there is also the exact expression. The solution to

y' = -a*y + b*exp(-t)          a = 100000  b = 99999

is

y = A*exp(-a*(t-t0)) + (b/(a-1))*exp(-t)

where A is determined by the initial condition y(t0) = y0

A = y0 - (b/(a-1))*exp(-t0)

In this case b/(a-1) = 1 (probably not by coincidence) so further simplification happens.