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## Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9

Choose the correct or the most suitable answer from the given four alternatives:

Question 1.

The order and degree of the differential equation are respectively ………

(a) 2, 3

(b) 3, 3

(c) 2, 6

(d) 2, 4

Solution:

(a) 2, 3

Hint:

Order = 2,

degree = 3

Question 2.

The differential equation representing the family of curves y = A cos (x + B), where A and B

are parameters, is …….

Solution:

(b) \(\frac{d^{2} y}{d x^{2}}+y=0\)

Hint:

Question 3.

The order and degree of the differential equation is …..

(a) 1, 2

(b) 2, 2

(c) 1, 1

(d) 2, 1

Solution:

(c) 1, 1

Hint:

Since, the first order derivatives are involved, Order = 1 and degree 1.

Question 4.

The order of the differential equation of all circles with centre at (h, k) and radius ‘a’ is …….

(a) 2

(b) 3

(c) 4

(d) 1

Solution:

(a) 2

Hint:

Equation of circle is (x – h)^{2} + (y – k)^{2} = a^{2}

Equation is to be differentiated twice as two parameters are given.

∴ Order = 2

Question 5.

The differential equation of the family of curves y = Ae^{x} + Be^{-x}, where A and B are arbitrary constants is ……

Solution:

\(\frac{d^{2} y}{d x^{2}}-y=0\)

Hint:

Question 6.

The general solution of the differential equation is …..

(a) xy = k

(b) y = k log x

(c) y = kx

(d) log y = kx

Solution:

(c) y = kx

Question 7.

The solution of the differential equation represents ……..

(a) straight lines

(b) circles

(c) parabola

(d) ellipse

Solution:

(c) parabola

Hint:

Question 8.

The solution of is …….

Solution:

(b) \(y=c e^{-\int \mathbf{P} d x}\)

Hint:

Question 9.

The integrating factor of the differential equation is …….

Solution:

(b) \(\frac{e^{x}}{x}\)

Hint:

Question 10.

The integrating factor of the differential equation is x, then P(x) …………

Hint:

Question 11.

The degree of the differential equation is ……..

(a) 2

(b) 3

(c) 1

(d) 4

Solution:

(c) 1

Hint:

Degree = 1

Question 12.

If p and q are the order and degree of the differential equation when …..

(a) p < q

(b) p = q

(c) p > q

(d) p exists and q does not exist

Solution:

(c) p > q

Hint:

Question 13.

The solution of the differential equation is …….

Solution:

(a) \(y+\sin ^{-1} x=c\)

Hint:

Question 14.

The solution of the differential equation \(\frac{d y}{d x}=2 x y\) is ………

Solution:

(a) \(y=c e^{x^{2}}\)

Hint:

Question 15.

The general solution of the differential equation is ……

Solution:

(b) \(e^{x}+e^{-y}=c\)

Hint:

Question 16.

Solution:

(c) \(\frac{1}{2^{x}}-\frac{1}{2^{y}}=c\)

Hint:

Question 17.

The solution of the differential equation is ……..

Solution:

(b) \(\phi\left(\frac{y}{x}\right)=k x\)

Hint:

Question 18.

If sin x is the integrating factor of the linear differential equation , then P is ……

(a) log sin x

(b) cos x

(c) tan x

(d) cot x

Solution:

(d) cot x

Hint:

Question 19.

The number of arbitrary constants in the general solutions of order n and n + 1 is respectively ……….

(a) n – 1, n

(b) n, n + 1

(c) n + 1, n + 2

(d) n + 1, n

Solution:

(b) n, n + 1

Question 20.

The number of arbitrary constants in the particular solution of a differential equation of third order is ………….

(a) 3

(b) 2

(c) 1

(d) 0

Solution:

(d) 0

Question 21.

The integrating factor of the differential equation is ……..

Solution:

(a) \(\frac{1}{x+1}\)

Hint:

Question 22.

The population P in any year t is such that the rate of increase in the population is proportional to the population. Then ……

Solution:

(a) \(\mathbf{P}=c e^{k t}\)

Hint:

Question 23.

P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaining, then ……

(a) P = ce^{kt}

(b) P = ce^{-kt}

(c) P = ckt

(d) Pt = c

Solution:

(b) P = ce^{-kt}

Hint:

Question 24.

(a) 2

(b) -2

(c )1

(d) -1

Solution:

(b) -2

Hint:

Question 25.

The slope at any point of a curve y =f (x) is given by and it passes through (-1, 1). Then the equation of the curve is ……..

(a) y = x^{3} + 2

(b) y = 3x^{2} + 4

(c) y = 3x^{3} + 4

(d) y = x^{3} + 5

Solution:

(a) y = x^{3} + 2

Hint:

### Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 Additional Problems

Choose the correct or the most suitable answer from the given four alternatives:

Question 1.

The integrating factor of is ………

(a) log x

(b) x^{2}

(c) e^{x}

(d) x

Solution:

(b) x^{2}

Hint:

Question 2.

If cos x is an integrating factor of the differential equation then P = ……

(a) – cot x

(b) cot x

(c) tan x

(d) – tan x

Solution:

(d) – tan x

Question 3.

The integrating factor of dx + x dy = e^{-y} sec^{2}y dy is ……….

(a) e^{x}

(b) e^{-x}

(c) e^{y}

(d) e^{-y}

Solution:

(c) e^{y}

Hint:

Question 4.

Integrating factor of

is …..

(a) e^{x}

(b) log x

(c) \(\frac{1}{x}\)

(d) e^{-x}

Solution:

(b) log x

Hint:

Question 5.

The solution of where m < 0 is ……

Solution:

\(x=c e^{-m y}\)

Hint:

Question 6.

y = cx – c^{2} is the general solution of the differential equation …….

Solution:

(a) \(\left(y^{\prime}\right)^{2}-x y^{\prime}+y=0\)

Hint:

Question 7.

The differential equation of all non-vertical lines in a plane is ……

Solution:

\(\frac{d^{2} y}{d x^{2}}=0\)

Hint:

The equation of the straight line is y = mx + c

Question 8.

The differential equation of all circles with center at the origin is

(a) x dy + y dx = 0

(b) x dy – y dx = 0

(c) x dx + y dy = 0

(d) x dx – y dy = 0

Solution:

(c) x dx + y dy = 0

Hint:

The equation of family of circle with the centre at the origin is x^{2} + y^{2} = a^{2}

Question 9.

The differential equation of the family of lines y = mx is ……..

Solution:

(d) 6

Hint:

Question 10.

The degree of the differential equation

(a) 1

(b) 2

(c) 3

(d) 6

Solution:

(d) 6

Hint:

Question 11.

(a) 1

(b) 3

(c) -2

(d) 2

Solution:

(b) 3

Hint:

Question 12.

The amount present in a radioactive element disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is (k is negative)

Solution:

(c) \(\frac{d p}{d t}=k p\)

Hint:

Let p be the amount present in a radio active element

Question 13.

On putting y = vx, the homogeneous differential equation x^{2}dy + y (x + y)dx = 0 becomes …….

(a) xdv + (2v + v^{2}) dx = 0

(b) vdx + (2x + x^{2})dv = 0

(c) v^{2}dx – (x + x^{2}) dv = 0

(d) vdv + (2x + x^{2}) dx = 0

Solution:

(a) xdv + (2v + v^{2}) dx = 0

Hint: