Page 154 - DMTH202_BASIC_MATHEMATICS_II

Unit 11: Linear Differential Equations of First Order

Notes
!
Caution If n = 0, Bernoulli’s equation decreases instantaneously to the standard form first-
order linear equation:

dy
 Py  Q
dx
(ii) General equation reducible to linear form is

dy
f ' ( )  Pf   y  Q …..(4)
y
dx
where P and Q are functions of x only or constants.
dy dz
Putting f (y) = z, so that ' ( ) .f y 
dx dx
Equation (4) becomes

dz
 Pz  Q
dx
which is linear.

Example:

dy x
Solve  2 y  x y …..(1)
dx 1 x

Solution:

Dividing by y , the equation (1) becomes

 1 dy x 1
y 2  2 y 2  x …..(2)

dx 1 x
Putting y 1 2  z

1  1 dy dz
So that y 2 
2 dx dx
 1 dy dz
or y 2  2 .
dx dx
Equation (2) becomes

dz x
2  z  x
dx 1 x 2

dz x x
or  2 z  , which is linear in z.

dx 2(1 x ) 2

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