Chapter 1

First Order Linear Differential Equations

OverviewCalculus

Differential Integral

Area under curve Slope

Calculus Relation

position velocity acceleration

Differential

Integral

Diff. Eq.

Diff. Eq.

Ordinary Partial

involving only oneindependent variableand derivatives-Chapter 1 and 2

involve more than oneindependent variableand partial derivatives-Chapter 5

Outline1. First Order Linear Differential Equations (8 Hours)

1.1Formation and solution of differential equation. 1.2Initial and boundary value problems. 1.3Methods of solution: (a) Separating the variables. (b) Homogeneous. (c) Linear. (d) Exact.

1.4Application of first order differential equation: (a)Growth population. (b)Newtons Law of cooling. (c)Linear motion. (d)Simple electric circuit

Order of an ordinary diff. eq.

Order 1

or

Order 2

or

Order n

or

Order of an ordinary diff. eq.

Example

State the order of the following diff. equation

a) + + = 0

b) + 4 + 3 =

c) 1 =

Formation of ordinary diff. eq.

Formation of ordinary diff. eq. by eliminating constant. Method :

1) Differentiate2) Multiply with x (if required)3) Add or subtract to eliminate constant

ExampleFind the differential equation by eliminating constant A and B in the following equation

1) = + 2) = +

Solution of ordinary diff. eq.

Example

Given that + 6 = 0. Show that

a) = 0is a solutionb) = 0 and = are solutionsc) = 5 + 4 is also a solutiond) = is not a solution

Solution of ordinary diff. eq.

ExampleShow that = + ( + 2) is a general solution for the differential equation

= ( + 3)

Hence find the value of A if y=4 when x=0

Initial and Boundary Value Problems

Initial conditions Condition which have the same value for the

independent variable. 0 = 0 and (0) = 2

Initial value problem Differential equations together with its initial equation Solve the equation

+ 6 + 8 = cosh 2Subject to initial condition

0 = 0 and (0) = 2

Initial and Boundary Value Problems, cont.

ExampleShow that

=

where A and B are constants, is a general solution of the equation

+ 2 + 9 = 0Which satisfied the initial condition

= 3 and ( ) = 0

Boundary Conditions and Boundary Value Problem

Boundary conditions Conditions which have different values for the

independent variable0 = 0 and 1 =2

Boundary value problem Differential equation together with its boundary

conditions Solve the equation

+ 3 + 2 = 3Which satisfied the boundary condition

0 = 0 and (1) =

Boundary Conditions and Boundary Value Problem, cont.

ExampleShow that

= +where A and B are constants, is a general solution of the equation

+ 9 = 0Which satisfied the initial condition

2 = 1 and (1) = 0