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 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 = 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 = 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 dififerential 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 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

Question 21.
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 = 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 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) x²
(c) e^x
(d) x
Solution:
(b) x²
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 sec²y 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.
Solution of where m < 0 is ……

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

Question 6.
y = cx – c² 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 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

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 radio active 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 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: