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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