# Solving ODE with Heaviside Step and Delta function

1. Oct 30, 2012

### dominic.tsy

1. The problem statement, all variables and given/known data

Find the solution of the equation:

α(dy/dt) + y = f(t)

for the following conditions:
(a) when f(t) = H(t) where H(t) is the Heaviside step function
(b) when f(t) = δ(t) where δ(t) is the delta function
(c) when f(t) = β^(-1)e^(t/β)H(t) with β<α

2. Relevant equations

3. The attempt at a solution

1.
α(dy/dt) + y = f(t)
2.
(dy/dt) + (1/α)y = (1/α)f(t)
3.
finding the integrating factor
μ(t) = e^(∫(1/α)dt) = e^(t/α)
4.
[e^(t/α)](dy/dt) + (1/α)[e^(t/α)]y = (1/α)[e^(t/α)]f(t)
5.
d/dt{[e^(t/α)]y}=[(e^(t/α))/α]f(t)
6.
∫d/dt{[e^(t/α)]y}dt=∫[(e^(t/α))/α]f(t)dt
7.
[e^(t/α)]y = ......

then i dont know how to continue

Do you know what those functions are? The 'Heaviside step function' is 0 if x< 0, 1 if $x\ge 0$. So $\int H(x)e^{-t/a} dx= 0$ if x is negative and $\int e^{-t/a} dt$ if x is positive.
The 'delta function' (strictly speaking not a function but a 'generalized function' or 'distribution') has the property that for any function, f(x), $\int f(x)\delta(x)dx= f(0)$.