Solve the integral equation

Question

Solve the integral equation

t

f(t) + ∫ f(Τ) dΤ =1

0

Note: the capital T's are the greek letter Tau I'm not sure if that makes much of a difference but I thought I ought to clarify.

Answer 1

Taking the derivative of both sides of the equation, we get f'(t) + f(t) = 0.

So, f'(t) = -f(t)

Letting y = f(t), we have dy/dt = -y

Separating variables, (1/y)dy = -dt

Integrate to get lnlyl = -t + C

lyl = e^-t+C = e^Ce^-t

y = Ce^-t

From the original equation, if t = 0, we see that f(0) = 1.

So, Ce⁰ = 1. Therefore, C = 1

Thus, f(t) = e^-t