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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.3

Question 1.

Find the differential equation of the family of

(i) all non-vertical lines in a plane

Solution:

Equation of family of all non-vertical lines is

y = mx + c (m ≠ 0)

Differentiate with respect to ‘x’

(ii) all non-horizontal lines in a plane.

Solution:

Equation of family of all non-horizontal lines is

Question 2.

Form the differential equation of all straight lines touching the circle x^{2} + y^{2} = r^{2}

Solution:

Equation of circle x^{2} + y^{2} = r^{2} of the line y = mx + c is to be a tangent to the circle, then the equation of the tangent is

Differentiating with respect to V dy

Question 3.

Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.

Solution:

All circles passing through the origin and having their centre on the x -axis say at (a, 0) will have radius ‘a’ units.

∴ Equation of circle is (x – a^{2}) + y^{2} = a^{2} ….(1) [ ∵ ‘a’ arbitrary constant]

Differentiate with respect to ‘x’

Question 4.

Find the differential equation of the family of all the parabolas with latus rectum 4a and whose axes are parallel to the x -axis.

Solution:

Equation of all parabolas whose axis is parallel to X – axis is

(y – k)^{2} = 4a (x – h)

Where (h, k) is the vertex

Question 5.

Find the differential equation of the family of parabolas with vertex at (0, -1) and having axis along the y – axis.

Solution:

Given, vertex (0, -1) and axis along y-axis

Equation of Parabola, (x + 1)^{2} = – 4ay …… (1) [∵ a is the perameter]

Differentiate with respect to ‘x’

Question 6.

Find the differential equations of the family of all the ellipses having foci on the y – axis and centre at the origin.

Solution:

Equations of the family of all the Ellipses having foci on the y – axis and centre at the origin is

Differentiate with respect to ’x’

Question 7.

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^{8x} + Be^{-8x}, where A and B are arbitrary constants.

Solution:

Question 8.

Find the differential equation of the curve represented by xy = ae^{x} + be^{-x} + x^{2}.

Solution:

xy = ae^{x} + be^{-x} + x^{2}

xy – x^{2} = ae^{x} + be^{-x} …… (1) [∵ a’, b’ are arbitrary constants]

Differentiate with respect to ‘x’

xy’ +y – 2x = ae^{x} – be^{-x}

Again, Differentiate with respect to ‘x’

### Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.3 Additional Problems

Question 1.

Find the differential equation of the family of straight lines y = mx + \(\frac{a}{m}\) when

(i) m is the parameter,

(ii) a is the parameter,

(iii) a, m both are parameters.

Solution:

Question 2.

Find the differential equation that will represent family of all circles having centres on the x-axis and the radius is unity.

Solution:

Equation of a circle with centre on x-axis and radius 1 unit is

(x – a)^{2} + y^{2} = 1 ….. (1)

Differentiating with respect to x,

2 (x – a) + 2yy’ – 0

⇒ 2 (x – a) = – 2yy’

(or) x – a = -yy’ ……(2)

Substituting (2) in (1), we get,

Question 3.

From the differential equation from the following equations.

(i) y = e^{2x} (A + Bx)

Solution:

ye^{-2x} = A + Bx ……. (1)

Since the above equation contains two arbitrary constants, differentiating twice,

(ii) y = e^{x}(A cos 3x + B sin 3x)

Solution:

(iii) Ax^{2} + By^{2} = 1

Solution:

Eliminating A and B between (1), (2) and (3) we get

(iv) y^{2} = 4a(x – a)

Solution: