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Differential Equation

9.1. Order of a Differential Equation.

Q.1. Write the order and degree of the following differential equation :

(dy/dx)⁴ + 3yd²y/dx² = 0.

Solution :

We have, (dy/dx)⁴ + 3yd²y/dx² = 0
Order of the differential equation = order of highest derivative = 2. [Ans.]

Degree of the differential equation = degree of highest derivative = 1. [Ans.]

Q.2. Write the order and degree of the differential equation :

(d²y/dx²)² + (dy/dx)³ + 2y = 0

Solution :

We have, (d²y/dx²)² + (dy/dx)³+ 2y = 0
Order = 2, Degree = 2. [Ans.]

9.2. General and Particular Solutions of a Differential Equation.

Q.1. Verify that y = A cos x – B sin x is a solution of the differential equation d²y/dx² + y = 0.

Solution :

We have, y = A cos x – B sin x -------------------- (1)
Differentiating we get, dy/dx = – A sin x – B cos x
Differentiating again we get, d²y/dx² = – A cos x + B sin x
= – (A cos x – B sin x)
= – y [Using (1)]
Or, d²y/dx² + y = 0.
Hence, (1) is a solution of the given differential equation. [Proved.]

9.3. Formation of Differential Equation.

Q.1. Form the differential equation of the family of curves y = a sin (x + b), where a and b are arbitrary constants.

Solution :

We have, y = a sin (x + b) --------------------- (1)
Differentiating both sides of (1) with respect to x, we get
dy/dx = a cos (x + b) ------------------ (2)
Differentiating both sides of (2) with respect to x, we get
d²y/dx² = – a sin (x + b) ------------------- (3)
= – y [Using (1)]
Or, d²y/dx² + y = 0
This is the required differential equation whose solution is (1).

Paper By Mr. M.P.Keshari
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