Notes On Differential Equations of Family of Curves - CBSE Class 12 Maths

Differential equations are used to depict some geometrical properties of curves.

Ex:

The equation of a circle whose centre is at the origin is x2 + y2 = r2

Differentiating with respect to x:

2x + 2y dy/dx = 0

⇒ x + y dy/dx = 0 [Required differential equation]

⇒ x = - y dy/dx ⇒ x dx = - y dy

Understanding how the differential equation represents family of circles:

⇒ ∫ x dx = - ∫ y dy

⇒ x2/2 = - y2/2 + C1

⇒ x2 = - y2 + 2C1

⇒ x2 + y2 = C, Where C =  2C1, C ∈ R

Summary:

1. The equation representing a family of curves with one unknown parameter:

F1(x,y,a) = 0 ... (1)

2. Differentiating F1(x,y,a) : g(x,y,y',a) = 0 ......(2)

3. Eliminating a between (1) and (2):

F(x,y,y') = 0 ......(3) [Required differential equation]

Observe that the differential equation obtained does not contain unknown parameter a, and contains the derivative.

The equation representing a family of curves with two unknown parameters:

F2(x,y,a,b) = 0 .....(1). Here, 'a' and 'b' are the unknown parameters.

2. Differentiating F2(x,y,a,b): g(x,y,y',a,b) = 0 .....(2)

Since there are two unknown parameters, a and b, two equations, one and two, are not sufficient to eliminate them.

3. Differentiating g(x,y,y',a,b): h(x,y,y',y",a,b) = 0 ......(3)

3. Eliminating a and b between (1), (2) and (3):

F(x,y,a,b) = 0 [Required differential equation]

Note: The number of unknown parameters determines the order of the differential equation. If the number of unknown parameters present in the equation corresponding to the family of curves is two, then the order of the differential equation obtained is two.

Summary

Differential equations are used to depict some geometrical properties of curves.

Ex:

The equation of a circle whose centre is at the origin is x2 + y2 = r2

Differentiating with respect to x:

2x + 2y dy/dx = 0

⇒ x + y dy/dx = 0 [Required differential equation]

⇒ x = - y dy/dx ⇒ x dx = - y dy

Understanding how the differential equation represents family of circles:

⇒ ∫ x dx = - ∫ y dy

⇒ x2/2 = - y2/2 + C1

⇒ x2 = - y2 + 2C1

⇒ x2 + y2 = C, Where C =  2C1, C ∈ R

Summary:

1. The equation representing a family of curves with one unknown parameter:

F1(x,y,a) = 0 ... (1)

2. Differentiating F1(x,y,a) : g(x,y,y',a) = 0 ......(2)

3. Eliminating a between (1) and (2):

F(x,y,y') = 0 ......(3) [Required differential equation]

Observe that the differential equation obtained does not contain unknown parameter a, and contains the derivative.

The equation representing a family of curves with two unknown parameters:

F2(x,y,a,b) = 0 .....(1). Here, 'a' and 'b' are the unknown parameters.

2. Differentiating F2(x,y,a,b): g(x,y,y',a,b) = 0 .....(2)

Since there are two unknown parameters, a and b, two equations, one and two, are not sufficient to eliminate them.

3. Differentiating g(x,y,y',a,b): h(x,y,y',y",a,b) = 0 ......(3)

3. Eliminating a and b between (1), (2) and (3):

F(x,y,a,b) = 0 [Required differential equation]

Note: The number of unknown parameters determines the order of the differential equation. If the number of unknown parameters present in the equation corresponding to the family of curves is two, then the order of the differential equation obtained is two.

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