FINDING ORDER AND DEGREE OF DIFFERENTIAL EQUATION

The order of a differential equation depends on the derivative of the highest order in the equation.

The degree of a differential equation, similarly, is determined by the highest exponent on any variables involved.

Find the order and degree of the following differential equations.

(1)  (dy/dx) + y  =  x2

(2)  y' + y2  =  x

(3)  y'' + 3 (y')2 + y3

(4)  d2y/dx2 + x  =  √[y + (dy/dx)]

(5)  d2y/dx2 - y + (dy/dx + d3y/dx3)(3/2)  =  0

(6)  y''  =  (y - (y')3)(2/3)

(7)  y' + (y'')2  =  (x + y'')2

(8)  (dy/dx)2 + x = (dx/dy) + x2

Problem 1 :

(dy/dx) + y  =  x2

Solution :

By differentiating y with respect to x, we get dy/dx and its highest exponent is 1. So, order  =  1 and degree  =  1.

Order  =  1,  Degree  =  1.

Problem 2 :

y' + y2  =  x

Solution :

By differentiating y with respect to x, we get y' and its highest exponent is 1. So, order  =  1 and degree  =  1.

Problem 3 :

y'' + (3y')2 + y3

Solution :

By differentiating y with respect to x two times, we get y''. So, order  =  2 and degree  =  1.

Problem 4 :

d2y/dx2 + x  =  √[y + (dy/dx)]

Solution :

d2y/dx2 + x  =  √[y + (dy/dx)]

Take squares on both sides.

(d2y/dx2 + x)2  =  [y + (dy/dx)]

By differentiating y with respect to x two times, we get d2y/dx2. So, order  =  2 and degree  =  2.

Problem 5 :

d2y/dx2 - y + (dy/dx + d3y/dx3)(3/2)  =  0

Solution :

d2y/dx2 - y + (dy/dx + d3y/dx3)(3/2)  =  0

d2y/dx2 - y  =  (dy/dx + d3y/dx3)(3/2)

Take squares on both sides

(d2y/dx2 - y)2  =  (dy/dx + d3y/dx3)3

So, order  =  3 and degree  =  3.

Problem 6 :

y''  =  (y - (y')3)(2/3)

Solution :

y''  =  (y - (y')3)(2/3)

Take cubes on both sides.

(y'')3  =  (y - (y')3)2

So, order is 2 and degree is 3.

Problem 7 :

y' + (y'')2  =  (x + y'')2

Solution :

y' + (y'')2  =  (x + y'')2

y' + (y'')2  =  x2 + (y'')2 + 2xy''

x2- 2xy''-y'  =  0

So, the order is 2 and degree is 1.

Problem 8 :

(dy/dx)2 + x  =  (dx/dy) + x2

Solution :

(dy/dx)2 + x  =  (dx/dy) + x2

(dy/dx)2 + x  =  [1/(dy/dx)] + x2

(dy/dx)3 + x(dy/dx)  =  =  1 + x2(dy/dx)

So, the order is 1 and degree is 3.

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