Chapter one
Some important basics of differential equations
Definition:
A differential equation is an equation involving derivatives or differentials.
For example
d 2y
dy

x
 y 0
(2) x
dx 2
dx
dy
(1)
y x
dx
2
2
3
d 3 y  d 2 y  dy
 2  
 x 2 1
(3)
3
dx
 dx  dx
 d 2 y   dy 
(4)  2      y  cos x
 dx   dx 
u
u
(5) x
y
 3u
x
y
 2u
 2u u
y

(6) x
x 2
y 2 x
4
Definition:
Ordinary differential equation is a differential equation involving only ordinary
derivatives with respect to a single independent variable.
For example equations (1), (2), (3) and (4).
Definition:
Partial differential equation is a differential equation involving partial derivatives
with respect to more than one independent variable.
For example equations (5) and (6).
Definition:
Order of a differential equation is that of the highest derivative (or differential) in it.
For example:
(i) Equations (1) and (5) are of order one.
(ii) Equations (2), (4) and (6) are of order two.
(iii) Equation (3) of order three.
1
Definition:
Degree of differential equation, which is algebraic in the derivatives (or differentials),
is the power of highest derivative (or differential) in it when the equation is free from
radicals and fractions in the dependent variable and its derivatives.
For example:
(i) Equations (1), (2), (3), (5) and (6) are of first degree.
(ii) Equation (4) is of third degree.
Example:
Find the order and degree of the following differential equation
2
 d 2y 
 dy
3
 dx 2   1   dx




.

Solution:
This equation can be written as
2
 d 2 y  3   dy
 dx 2   1   dx

  
1
2
 d 2 y    dy
 2


 dx 2   1   dx


  
3
 2
 

4
 d 2 y    dy
 dx 2   1   dx

  



3
Therefore, this equation is of second order and fourth degree.
Definition:
Solution of an n th-order differentiate equation is a function which is n times
differentiable and satisfies the differential equation.
For example, y  sin x is solution for the equation y   y  0 , where
y  sin x  y   cos x  y    sin x
 y   y   sin x  sin x  0 .
Definition:
General solution of differentiate equation is the solution which contains arbitrary
constants and the number of these constants equal to the order of the differential
equation.
For example, y   0 has solution y  c1x 2  c 2x  c 3 .
2
Conformation of differential equation:
Let
f ( x , y ,c )  0
(I)
express a family of curves, where c is constant. Differentiate equation (1), we get new
equation
F (x , y , y ,c )  0
(II)
eliminate the constant c from equations (I) and (II), we obtain the equation
g (x , y , y )  0
(III)
which is the differential equation correspondence to equation (I).
Example:
Find the differential equations which correspond to the following families of curves
(1) y  c1x  c 2 , c1 , c 2 are arbitrary constants.
(2) y  A cos x  B sin x ,
A , B are arbitrary constants.
(3) y  cx 2  c 2 , c is an arbitrary constant.
(4) y  c1 e 2 x  c 2 e x  c 3 , c1 , c 2 , c 3 are arbitrary constants.
Solution:
(1) y  c1x  c 2
Since the equation contains two constants, then we differentiate this equation twice.
y  c1x  c 2  y   c1  y   0
y   0 is the differential equation correspondence to the equation y  c1x  c 2 .
(2) y  A cos x  B sin x  y   A sin x  B cos x  y    A cos x  B sin x
 y   A cos x  B sin x    A cos x  B sin x    y  y   y  0
y   y  0 is the differential equation correspondence to the equation
y  A cos x  B sin x .
3
(3) y  cx 2  c 2
Here, we differentiate this equation once only because this equation contains one
arbitrary constant c
y
y   2cx  c 
2x
y 2  y 
y 
x 

2x
 2x 
2
1
1
multiply by 4 x
 4x 2 y  2x 3 y   y 2
 y  xy   2 y 2 
2
4x
2
 y 2  2x 3 y   4x 2 y  0 is the differential equation correspondence to the equation
y  cx 2  c 2 .
(4) y  c1 e 2 x  c 2 e x  c 3
y   2c1 e 2 x  c 2 e x
e  x y   2c1 e x  c 2
 y   2c1 e 2 x  c 2 e x
multiply by e  x
differentiate with respect to x
e  x y   e  x y   2c1 e x  e  x ( y   y )  2c1 e x
e 2 x ( y   y )  2c1
differentiate with respect to x
e 2 x ( y   y )  2e 2 x ( y   y )  0
( y   y )  2( y   y )  0
multiply by e  x
multiply by e 2x
 y   3 y   2 y   0
y   3 y   2 y   0 is the differential equation correspondence to the equation
y  c1 e 2 x  c 2 e x  c 3 .
4
Exercises
(I) Find the order and degree of the following differential equations
2
 dy  3x
(1)   
 dx  4 y
d 2y
dy

2
x
y x2 2
(2) x
2
dx
dx
2
2
d 2y
 d 3y 
 dy 
(3)  3   2 2  x 2    0
dx
 dx 
 dx 
  dy 2 
(4) 1    
  dx  
3
3l 2
d 2y
k
dx 2
  dy 2 
 d 2y 
(5)  2   k 1    
 dx 
  dx  
1/ 3
5l 2
(II) Find the differential equations which correspond to the following families of the
curves
(1) y  x 3  c
(2) y  c1x 2  c 2
(3) (x  c )2  y 2  c 2
(4) y  c1 e x  c 2 x
(5) y  c1 e x  c 2 e  x
(6) y  c1 sin(x  c 2 )
(7) y  e x (c1 sin 2x  c 2 cos2x )
(8) y  c1  c 2e x  sin x
5