Differential Equation

6 )

If m and n are order and degree of the equation (d²y / dx²)⁵ + 4[(d²y / dx²)³ / (d³y / dx³)] + (d³y / dx³) = x² – 1, then:

m = 3, n = 2

m = 3, n = 3

m = 3, n = 5

m = 3, n = 1

View Solution

Answer :

(d³y / dx³) ∙ (d²y / dx²)⁵ + 4(d²y / dx²)³ + (d³y / dx³)² = (x² – 1) (d³y / dx³)

Hence order = m = 3

degree = n = 2

7 )

The degree and order of the differential equation of the family of all parabolas whose axis is x-axis, are respectively.

1, 2

3, 2

2, 3

2, 1

View Solution

Answer :

Family of parabola with axis as x axis is y² = 4a(x + b)

∴ 2y(dy / dx) = 4a

∴ y(d²y / dx²) + (dy / dx)² = 0

∴ degree = 1 and order = 2

8 )

The differential equation representing the family of curves y² = 2c(x + √c), where c is a positive parameter, is of order and degree as follows.

order 2, degree 1

order 1, degree 2

order 2, degree 2

order 1, degree 3

View Solution

Answer :

y² = 2c(x + √c) (1)

∴ 2yy¹ = 2c

∴ c = yy¹

∴ from (1), y² = 2yy¹[x + √(yy¹)]

∴ (y² – 2yy¹x)² = (yy¹)³ × 4

i.e. (y² – 2yy¹x)² = 4(yy¹)³

i.e. degree = 3

order = 1

9 )

The differential equation whose solution is Ax² + By² = 1, where A and B are arbitrary constants is of.

second order and second degree

first order and first degree

first order and second degree

second order and first degree

View Solution

Answer :

given: Ax² + Bx² = 1

differentiating, we get

2Ax + 2By(dy / dx) = 0

2A + 2B[(dy / dx)² + y(d²y / dx²)] = 0

∴ – [(2By) / x] (dy / dx) + 2B[(dy / dx)² + y ∙ (d²y / dx²)] = 0

∴ y ∙ (d²y / dx²) + (dy / dx)² – (y/x)(dy / dx) = 0

∴ so order = 2

degree = 1

10 )

Order and degree of differential equation of all tangent lines to the parabola y² = 4ax is_____

2, 2

3, 1

1, 2

4, 1

View Solution

Answer :

Equation of all tangent lines to parabola

y² = 4ax is y = mx + (a/m) (1)

where a = constant

m = arbitrary constant

∴ (dy / dx) = m

∴ from (1),

y = x(dy / dx) + [a / (dy / dx)]

∴ y ∙ (dy / dx) = x(dy / dx)² + a

∴ Equation is

(dy / dx)² – (y/x)(dy / dx) + (a/y) = 0

∴ order = 1

degree = 2

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