o Linear ordinary differential equations (ODE's) may be solved using Laplace ... The algebraic equation is solved for the Laplace transform of the sol...

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ME 3600 Control Systems Solutions to Linear, Ordinary Differential Equations Using Laplace Transforms o Linear ordinary differential equations (ODE's) may be solved using Laplace transforms. There are three steps in the solution process. 1. The Laplace transform is used to convert the differential equation into an algebraic equation. 2. The algebraic equation is solved for the Laplace transform of the solution, and the concept of partial fractions is used to separate the solution into its elementary components. 3. Finally, the inverse Laplace transform is taken of each of the elementary components to find the solution in the time domain. o The definitions of the Laplace transform and the inverse Laplace transform are given below on page 2. Fortunately, however, tables of known transforms can be used to solve many differential equations. o For most cases, the properties of the Laplace transform and tables of transforms of common functions are used to find solutions. A list of helpful properties of Laplace transforms is given below.

Properties of the Laplace Transform and Inverse Laplace Transforms 1.

(kf (t )) k (f (t )) kF (s)

and

2.

(f1 (t ) f 2 (t )) F1 (s) F2 (s)

and

3.

df dt

4.

t F (s ) f ( t ) dt s 0

sF (s) f (0)

and

1

(kF (s)) kf (t )

1

(F1 (s) F2 (s)) f1 (t ) f 2 (t )

d2 f 2 df s F ( s ) s f (0) (0) 2 dt dt

5. Initial Value Theorem: lim f (t ) lim s F ( s )

(if the time limit exists)

6. Final Value Theorem: lim f (t ) lim s F ( s )

(if the time limit exists)

t 0

t

s s 0

Note: Item 3 shows the formulae for the Laplace transforms of the first two derivatives of a function. There is a more general formula for the Laplace transform of the nth derivative of a function that is used for differential equations of order three or higher. Kamman – ME 3600 – page: 1/2

Definitions of the Laplace Transform and Inverse Laplace Transform The Laplace transform of a function f (t ) is defined as

f (t ) F (s) f (t ) e st dt 0

and the inverse Laplace transform is defined to be 1

F (s) f (t )

1

j

2 j

F (s) e st ds

j

where s is a complex-valued variable in the Laplace domain.

Kamman – ME 3600 – page: 2/2