Solving ODE with Heaviside Step and Delta function

In summary: C8.y(t) = [e^(-t/α)]∫[(e^(t/α))/α]δ(t)dt + Ce^(-t/α)(c)α(dy/dt) + y = (β^(-1))e^(t/β)H(t)2.(dy/dt) + (1/α)y = (1/α)(β^(-1))e^(t/β)H(t)3.finding the integrating factorμ(t) = e^(∫(
  • #1
dominic.tsy
6
0
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

My try for all 3 are as follow:

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 don't know how to continue

please help guys...thanks
 
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  • #2
(a)α(dy/dt) + y = H(t)2.(dy/dt) + (1/α)y = (1/α)H(t)3.finding the integrating factorμ(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)]H(t)5.d/dt{[e^(t/α)]y}=[(e^(t/α))/α]H(t)6.∫d/dt{[e^(t/α)]y}dt=∫[(e^(t/α))/α]H(t)dt7.[e^(t/α)]y = ∫[(e^(t/α))/α]H(t)dt + C8.y(t) = [e^(-t/α)]∫[(e^(t/α))/α]H(t)dt + Ce^(-t/α)(b)α(dy/dt) + y = δ(t)2.(dy/dt) + (1/α)y = (1/α)δ(t)3.finding the integrating factorμ(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)]δ(t)5.d/dt{[e^(t/α)]y}=[(e^(t/α))/α]δ(t)6.∫d/dt{[e^(t/α)]y}dt=
 

What is a Heaviside Step function and how is it used in solving ODEs?

The Heaviside Step function, also known as the unit step function, is a mathematical function that is defined as 0 for negative input and 1 for positive input. In solving ODEs, it is used to model sudden changes or discontinuities in a system.

What is a Delta function and how is it used in solving ODEs?

The Delta function, also known as the Dirac delta function, is a mathematical function that is defined as 0 for all points except at a single point, where it is infinity. In solving ODEs, it is used to model impulses or singularities in a system.

How are Heaviside Step and Delta functions used together in solving ODEs?

In solving ODEs, Heaviside Step and Delta functions are used together to model systems that have both sudden changes and impulses. The Heaviside Step function is used to represent the sudden changes, while the Delta function is used to represent the impulses.

What are some common applications of using Heaviside Step and Delta functions in solving ODEs?

Some common applications of using Heaviside Step and Delta functions in solving ODEs include modeling electrical circuits, control systems, and physical systems with sudden changes or impulses.

Are there any limitations to using Heaviside Step and Delta functions in solving ODEs?

Yes, there are limitations to using Heaviside Step and Delta functions in solving ODEs. These functions are idealizations and may not accurately represent real-world systems. Additionally, their use may lead to non-physical solutions in certain cases.

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