Ordinary Differential Equations:Cheat Sheet/Few Useful Definitions

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Wronskian of Two Functions[edit | edit source]

Definition[edit | edit source]

The Wronskian of two functions is given by

Useful Facts[edit | edit source]

  • If two functions are linearly dependent on an interval, then their Wronskian vanishes on that interval.

Laplace Transforms[edit | edit source]

Definition[edit | edit source]

The Laplace transform of at a complex number is

Properties[edit | edit source]

  • Linearity:

If then:

  • for

Laplace Transform of Few Simple Functions[edit | edit source]

Convolution[edit | edit source]

Definition[edit | edit source]

The convolution of and is

Properties[edit | edit source]

Convolution is:

  1. Associative
  2. Commutative
  3. Distributive over addition

First Order Ordinary Differential Equations