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

   

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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 Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 1 are respectively ………
(a) 2, 3
(b) 3, 3
(c) 2, 6
(d) 2, 4
Solution:
(a) 2, 3
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 121
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 …….
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 2
Solution:
(b) \(\frac{d^{2} y}{d x^{2}}+y=0\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 3

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

Question 3.
The order and degree of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 513 is …..
(a) 1, 2
(b) 2, 2
(c) 1, 1
(d) 2, 1
Solution:
(c) 1, 1
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 514
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 = a2
Equation is to be differentiated twice as two parameters are given.
∴ Order = 2

Question 5.
The differential equation of the family of curves y = Aex + Be-x, where A and B are arbitrary constants is ……
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 5
Solution:
\(\frac{d^{2} y}{d x^{2}}-y=0\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 6

Question 6.
The general solution of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 7 is …..
(a) xy = k
(b) y = k log x
(c) y = kx
(d) log y = kx
Solution:
(c) y = kx

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

Question 7.
The solution of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 8 represents ……..
(a) straight lines
(b) circles
(c) parabola
(d) ellipse
Solution:
(c) parabola
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 9

SamacheerKalvi.Guru

Question 8.
The solution of Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 10 is …….
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 100
Solution:
(b) \(y=c e^{-\int \mathbf{P} d x}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 11
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 12

Question 9.
The integrating factor of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 13 is …….
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 133
Solution:
(b) \(\frac{e^{x}}{x}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 14

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

Question 10.
The integrating factor of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 15 is x, then P(x) …………
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 155
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 515

Question 11.
The degree of the dififerential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 516 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 Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 517 when …..
(a) p < q
(b) p = q
(c) p > q
(d) p exists and q does not exist
Solution:
(c) p > q
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 177

Question 13.
The solution of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 18 is …….
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 188
Solution:
(a) \(y+\sin ^{-1} x=c\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 19

Question 14.
The solution of the differential equation \(\frac{d y}{d x}=2 x y\) is ………
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 518
Solution:
(a) \(y=c e^{x^{2}}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 21

Question 15.
The general solution of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 22 is ……
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 222
Solution:
(b) \(e^{x}+e^{-y}=c\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 23

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

Question 16.
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 24
Solution:
(c) \(\frac{1}{2^{x}}-\frac{1}{2^{y}}=c\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 25

Question 17.
The solution of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 26 is ……..
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 266
Solution:
(b) \(\phi\left(\frac{y}{x}\right)=k x\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 27

Question 18.
If sin x is the integrating factor of the linear differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 28, then P is ……
(a) log sin x
(b) cos x
(c) tan x
(d) cot x
Solution:
(d) cot x
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 29

Question 19.
The number of arbitrary constants in the general solutions of order n and n + 1 are 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

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

Question 21.
Integrating factor of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 299 is ……..
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 291
Solution:
(a) \(\frac{1}{x+1}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 30
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 31

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 ……
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 32
Solution:
(a) \(\mathbf{P}=c e^{k t}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 33

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 = cekt
(b) P = ce-kt
(c) P = ckt
(d) Pt = c
Solution:
(b) P = ce-kt
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 34

Question 24.
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 35
(a) 2
(b) -2
(c )1
(d) -1
Solution:
(b) -2
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 36

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

Question 25.
The slope at any point of a curve y =f (x) is given by Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 37 and it passes through (-1, 1). Then the equation of the curve is ……..
(a) y = x3 + 2
(b) y = 3x2 + 4
(c) y = 3x3 + 4
(d) y = x3 + 5
Solution:
(a) y = x3 + 2
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 377
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 38

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 Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 39 is ………
(a) log x
(b) x2
(c) ex
(d) x
Solution:
(b) x2
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 40

Question 2.
If cos x is an integrating factor of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 400 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 sec2y dy is ……….
(a) ex
(b) e-x
(c) ey
(d) e-y
Solution:
(c) ey
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 401

Question 4.
Integrating factor of Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 42
is …..
(a) ex
(b) log x
(c) \(\frac{1}{x}\)
(d) e-x
Solution:
(b) log x
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 422

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

Question 5.
Solution of Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 43 where m < 0 is ……
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 433
Solution:
\(x=c e^{-m y}\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 44

Question 6.
y = cx – c2 is the general solution of the differential equation …….
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 45
Solution:
(a) \(\left(y^{\prime}\right)^{2}-x y^{\prime}+y=0\)
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 46

Question 7.
The differential equation of all non-vertical lines in a plane is ……
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 47
Solution:
\(\frac{d^{2} y}{d x^{2}}=0\)
Hint:
The equation of the straight line is y = mx + c
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 48

Question 8.
The differential equation of all circles with centre 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 x2 + y2 = a2
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 49

Question 9.
The differential equation of the family of lines y = mx is ……..
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 50
Solution:
(d) 6
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 51

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

Question 10.
The degree of the differential equation Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 511
(a) 1
(b) 2
(c) 3
(d) 6
Solution:
(d) 6
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 512

Question 11.
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 53
(a) 1
(b) 3
(c) -2
(d) 2
Solution:
(b) 3
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 54

Question 12.
The amount present in a radio active element disintegrates at a rate proportional to its amount. The differential equation corresponding to the above statement is (k is negative)
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 55
Solution:
(c) \(\frac{d p}{d t}=k p\)
Hint:
Let p be the amount present in a radio active element
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 56

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

Question 13.
On putting y = vx, the homogeneous differential equation x2dy + y (x + y)dx = 0 becomes …….
(a) xdv + (2v + v2) dx = 0
(b) vdx + (2x + x2)dv = 0
(c) v2dx – (x + x2) dv = 0
(d) vdv + (2x + x2) dx = 0
Solution:
(a) xdv + (2v + v2) dx = 0
Hint:
Samacheer Kalvi 12th Maths Solutions Chapter 10 Ordinary Differential Equations Ex 10.9 57