Visual SolutionEasy

Question

The order and degree of the differential equation

\left(\frac{d^2y}{dx^2}\right)^3 = \left(1 + \left(\frac{dy}{dx}\right)^2\right)^{5/2}

are respectively:

(A)

2

and

3

(B)

2

and

6

(C)

2

and

5

(D)

3

and

2

Solution Path

Order = 2 (highest derivative). Square both sides to remove fractional exponent, degree = 6.

01Question Setup

1/4

Find the order and degree of

\left(\frac{d^2y}{dx^2}\right)^3 = \left(1 + \left(\frac{dy}{dx}\right)^2\right)^{5/2}

Order

= \;?

, Degree

= \;?

02Identify Order

2/4

Highest derivative is

\frac{d^2y}{dx^2}

, so order

= 2

. But the RHS has a fractional exponent

5/2

- degree is not defined until we remove it.

Order

= 2

03Remove Fractional ExponentKEY INSIGHT

3/4

Square both sides:

\left(\frac{d^2y}{dx^2}\right)^6 = \left(1 + \left(\frac{dy}{dx}\right)^2\right)^5

. Now degree

= 6

(power of highest order derivative).

Degree

= 6

04Final Answer

4/4

Order

= 2

, Degree

= 6

. Sum

= 8

\boxed{\text{Order } 2, \text{ Degree } 6}

Concepts from this question1 concepts unlocked

★

STANDARD

Differentiate the given family of curves n times and eliminate all n arbitrary constants to obtain the differential equation

n \text{ arbitrary constants} \implies n \text{ differentiations to eliminate}

Tested frequently in JEE for finding the order of a differential equation and connecting curve families to their ODEs

Order and degree problemsFamily of curvesEnvelope problems

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