MATH 20D Lecture 27

Example. Solve the IVP $y^{\prime \prime }+y=4u(t-3),y(0)=0,y^{\prime }(0)=1$

7.9 - Dirac Delta Function

Dirac Delta Function

The Dirac Delta Function or unit impulse function, denoted , is defined by two properties:

Dirac Delta function can sometimes be seen as having value at . In addition, the impulse can occur at an arbitraty time . In this case, we have

Fact:

1. where is the unit step (Heaviside) function

Properties of Dirac Delta Function

1. Let be continuous function such that , and let . Then
   
2. Laplace transforms for Dirac Delta functions:
   

Example. Solve the IVP: $2y^{\prime \prime }+y^{\prime }+2y=\delta (t-5);y(0)=y^{\prime }(0)=0$

Do the rest of it normally

Example. Solve the IVP: $y^{\prime \prime }+6y^{\prime }+5y=\delta (t)+\delta (t-2);y(0)=1,y^{\prime }(0)=0$

Do the rest normally