Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.6

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Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.6

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Question 1.
The degree of the differential equation \(\frac{d^{4} y}{d x^{4}}-\left(\frac{d^{2} y}{d x^{2}}\right)^{4}+\frac{d y}{d x}=3\) is _________
(a) 1
(b) 2
(c) 3
(d) 4
Answer:
(a) 1
Hint:
Since the power of \(\frac{d^{4} y}{d x^{4}}\) is 1

Question 2.
The order and degree of the differential equation \(\sqrt{\frac{d^{2} y}{d x^{2}}}=\sqrt{\frac{d y}{d x}+5}\) are respectively
(a) 2 and 3
(b) 3 and 2
(c) 2 and 1
(d) 2 and 2
Answer:
(c) 2 and 1
Hint:
Squaring both sides, we get \(\frac{d^{2} y}{d x^{2}}=\frac{d y}{d x}+5\)
So order = 2, degree = 1

Question 3.
The order and degree of the differential equation \(\left(\frac{d^{2} y}{d x^{2}}\right)^{\frac{3}{2}}-\sqrt{\left(\frac{d y}{d x}\right)}-4=0\) are respectively.
(a) 2 and 6
(b) 3 and 6
(c) 1 and 4
(d) 2 and 4
Answer:
(a) 2 and 6
Hint:

Question 4.
The differential equation \(\left(\frac{d x}{d y}\right)^{3}+2 y^{\frac{1}{2}}=x\) is _________
(a) of order 2 and degree 1
(b) of order 1 and degree 3
(c) of order 1 and degree 6
(d) of order 1 and degree 2
Answer:
(b) of order 1 and degree 3

Question 5.
The differential equation formed by eliminating a and b from y = ae^x + be^-x is _______
(a) \(\frac{d^{2} y}{d x^{2}}-y=0\)
(b) \(\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0\)
(c) \(\frac{d^{2} y}{d x^{2}}=0\)
(d) \(\frac{d^{2} y}{d x^{2}}-x=0\)
Answer:
(a) \(\frac{d^{2} y}{d x^{2}}-y=0\)
Hint:

Question 6.
If y = cx + c – c³ then its differential equation is ______

Answer:
(a) \(y=x \frac{d y}{d x}+\frac{d y}{d x}-\left(\frac{d y}{d x}\right)^{3}\)

Question 7.
The integrating factor of the differential equation \(\frac{d x}{d y}\) + Px = Q is _____
(a) e^∫Pdx
(b) ∫Pdx
(c) ∫Pdy
(d) e^∫Pdy
Answer:
(d) e^∫Pdy

Question 8.
The complementary function of (D² + 4) y = e^2x is _______
(a) (Ax + B) e^2x
(b) (Ax + B) e^-2x
(c) A cos 2x + B sin 2x
(d) Ax^-2x + Be^2x
Answer:
(c) A cos 2x + B sin 2x
Hint:
A.E = m² + 4 = 0 ⇒ m = ±2i
C.F = e^0x (A cos 2x + B sin 2x)

Question 9.
The differential equation of y = mx + c is (m and c are arbitrary constants).
(a) \(\frac{d^{2} y}{d x^{2}}=0\)
(b) y = x \(\frac{d y}{d x}\)
(c) x dy + y dx = 0
(d) y dx – x dy = 0
Answer:
(a) \(\frac{d^{2} y}{d x^{2}}=0\)

Question 10.
The particular integral of the differential equation \(\frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}+16 y=2 e^{4 x}\) is ________
(a) \(\frac{x^{2} e^{4 x}}{2 !}\)
(b) \(\frac{e^{4 x}}{2 !}\)
(c) x² e^4x
(d) xe^4x
Answer:
(c) x² e^4x
Hint:

Question 11.
Solution of \(\frac{d x}{d y}\) + px = 0
(a) x = ce^py
(b) x = ce^-py
(c) x = py + c
(d) x = cy
Answer:
(b) x = ce^-py
Hint:

Question 12.
If sec² x is an integrating factor of the differential equation \(\frac{d y}{d x}\) + Py = Q then P = _____
(a) 2 tan x
(b) sec x
(c) cos² x
(d) tan² x
Answer:
(a) 2 tan x
Hint:

Question 13.
The integrating factor of x\(\frac{d y}{d x}\) – y = x² is _____
(a) \(\frac{-1}{x}\)
(b) \(\frac{1}{x}\)
(c) log x
(d) x
Answer:
(b) \(\frac{1}{x}\)
Hint:

Question 14.
The solution of the differential equation \(\frac{d y}{d x}\) + Py = Q where P and Q are the function of x is ______

Answer:
(c) \(y e^{\int p d x}=\int \mathrm{Q} e^{\int p d x} d x+c\)

Question 15.
The differential equation formed by eliminating A and B from y = e^-2x (A cos x + B sin x) is _______
(a) y₂ – 4y₁ + 5y = 0
(b) y₂ + 4y₁ – 5y = 0
(c) y₂ – 4y₁ – 5y = 0
(d) y₂ + 4y₁ + 5y = 0
Answer:
(d) y₂ + 4y₁ + 5y = 0
Hint:

Question 16.
The particular integral of the differential equation f(D) y = e^ax where f(D) = (D – a)² ________
(a) \(\frac{x^{2}}{2} e^{2 x}\)
(b) xe^ax
(c) \(\frac{x}{2} e^{2 x}\)
(d) x² e^2x
Answer:
(a) \(\frac{x^{2}}{2} e^{2 x}\)

Question 17.
The differential equation of x² + y² = a² is _____
(a) x dy + y dx = 0
(b) y dx – x dy = 0
(c) x dx – y dx = 0
(d) x dx + y dy = 0
Answer:
(d) x dx + y dy = 0
Hint:
x² + y² = a²
⇒ 2x + 2y \(\frac{d y}{d x}\) = 0
⇒ x dx + y dy = 0

Question 18.
The complementary function of \(\frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0\) is ______
(a) A + B e^x
(b) (A + B) e^x
(c) (Ax + B) e^x
(d) (Ae^x + B)
Answer:
(a) A + B e^x
Hint:
A.E is m² – m = 0
⇒ m(m – 1) = 0
⇒ m = 0, 1
CF is Ae^0x + Be^x = A + Be^x

Question 19.
The P.I of (3D² + D – 14)y = 13e^2x is _______
(a) \(\frac{x}{2}\) e^2x
(b) x e^2x
(c) \(\frac{x^{2}}{2}\) e^2x
(d) 13xe^2x
Answer:
(b) xe^2x
Hint:

Question 20.
The general solution of the differential equation \(\frac{d y}{d x}\) = cos x is _______
(a) y = sin x + 1
(b) y = sin x – 2
(c) y = cos x + c, c is an arbitrary constant
(d) y = sin x + c, c is an arbitrary constant
Answer:
(d) y = sin x + c, c is an arbitrary constant

Question 21.
A homogeneous differential equation of the form \(\frac{d y}{d x}=f\left(\frac{y}{x}\right)\) can be solved by making substitution
(a) y = vx
(b) v = yx
(c) x = vy
(d) x = v
Answer:
(a) y = vx

Question 22.
A homogeneous differential equation of the form \(\frac{d x}{d y}=f\left(\frac{x}{y}\right)\) can be solved by making substitution,
(a) x = vy
(b) y = vx
(c) y = v
(d) x = v
Answer:
(a) x = vy

Question 23.

Answer:
(d) \(\frac{1+v}{2 v^{2}} d v=-\frac{d x}{x}\)
Hint:

Question 24.
Which of the following is the homogeneous differential equation?
(a) (3x – 5) dx = (4y – 1) dy
(b) xy dx – (x³ + y³) dy = 0
(c) y² dx + (x² – xy – y²) dy = 0
(d) (x² + y) dx = (y² + x) dy
Answer:
(c) y² dx + (x² – xy – y²) dy = 0

Question 25.
The solution of the differential equation \(\frac{d y}{d x}=\frac{y}{x}+\frac{f\left(\frac{y}{x}\right)}{f^{\prime}\left(\frac{y}{x}\right)}\) is ______
(a) f(\(\frac{y}{x}\)) = kx
(b) x f(\(\frac{y}{x}\)) = k
(c) f(\(\frac{y}{x}\)) = ky
(d) y f(\(\frac{y}{x}\)) = k
Answer:
(a) f(\(\frac{y}{x}\)) = kx
Hint: