Page 160 - DMTH202_BASIC_MATHEMATICS_II

Unit 11: Linear Differential Equations of First Order

 To locate one quantity from the other you multiply the first by some number, then add a Notes
different number to the outcome.

 On integration, we get ye  Pdx   Qe  Pdx dx , c as the required general solution.

 Linear differential equation is commonly known as Leibnitz’s linear equation.
 The system is said to be consistent if it contains a solution. Or else the system is said to be
inconsistent.

 When concerning with two or more equations, it is fortunate to have a methodical technique
of determining if the system is consistent and to discover all solutions.

dy n
 An equation of the form  Py  Qy , where P and Q are functions of x only or constants
dx
is known as Bernoulli’s equation which can be made linear.

11.4 Keywords

dy n
Bernoulli’s Equation: An equation of the form  Py  Qy , where P and Q are functions of x
dx
only or constants is known as Bernoulli’s equation.

dy
Linear Equation: An equation of the form  Py  Q , in which P & Q are functions of x alone
dx
or constant is called a linear equation of the first order.

11.5 Review Questions

Solve the following differential equations:

3 dy
y
1. (x  2 )  y
dx
d  dy 2y 
2.     0
dx dx x 


dy 2 ( 3/2) x 1 x 2


3.  (1 x ) y  2 2
dx (1 x )

dy e x
4.  y  e
dx

dy 2
x
5. cosh x  y sinh x  2cosh sinh x
dx
dy  2  1  1 
x
6.    y   sin  2 
dx  x  x  x 

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