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What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge 2 when x = loge 2 and y = (4/3) loge3 when x = loge3?
- a)2
- b)-2
- c)-4
- d)4
Correct answer is option 'C'. Can you explain this answer?
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What will be the value of C if C the constant of the coefficient of th...
Solution:
Given equation: (D - 1)^2y = 0
Given values: y = 2loge2 when x = loge2, and y = (4/3)loge3 when x = loge3
We need to find the value of C, which represents the constant coefficient of the solution.
We can solve this problem by substituting the given values into the equation and solving for C.
Substituting the first set of values:
(1 - 1)^2y = 0
(0)^2y = 0
0 = 0
This equation is true for any value of y, which means that it is satisfied by the given values of y = 2loge2 when x = loge2.
Substituting the second set of values:
(3 - 1)^2y = 0
(2)^2y = 0
4y = 0
This equation is only satisfied when y = 0. However, the given value of y = (4/3)loge3 does not equal 0 when x = loge3. Therefore, this set of values does not satisfy the equation.
Since the equation is satisfied by the first set of values but not the second set, the value of C must be such that it cancels out the equation when y = 0.
Therefore, the value of C is -4, which is the coefficient that makes the equation (D - 1)^2y = 0 true when y = 0.
Hence, the correct answer is option C, -4.
Given equation: (D - 1)^2y = 0
Given values: y = 2loge2 when x = loge2, and y = (4/3)loge3 when x = loge3
We need to find the value of C, which represents the constant coefficient of the solution.
We can solve this problem by substituting the given values into the equation and solving for C.
Substituting the first set of values:
(1 - 1)^2y = 0
(0)^2y = 0
0 = 0
This equation is true for any value of y, which means that it is satisfied by the given values of y = 2loge2 when x = loge2.
Substituting the second set of values:
(3 - 1)^2y = 0
(2)^2y = 0
4y = 0
This equation is only satisfied when y = 0. However, the given value of y = (4/3)loge3 does not equal 0 when x = loge3. Therefore, this set of values does not satisfy the equation.
Since the equation is satisfied by the first set of values but not the second set, the value of C must be such that it cancels out the equation when y = 0.
Therefore, the value of C is -4, which is the coefficient that makes the equation (D - 1)^2y = 0 true when y = 0.
Hence, the correct answer is option C, -4.
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What will be the value of C if C the constant of the coefficient of th...
(D + 1)2y = 0
Or, (D2 + 2D+ 1)y = 0
⇒ d2y/dx2 + 2dy/dx + y = 0 ……….(1)
Let y = emx be a trial solution of equation (1). Then,
⇒ dy/dx = memx and d2y/dx2 = m2emx
Clearly, y = emx will satisfy equation (1). Hence, we have
⇒ m2.emx + 2m.emx + emx = 0
Or, m2 + 2m + 1 = 0 (as, emx ≠ 0) ………..(2)
Or, (m + 1)2 = 0
⇒ m = -1, -1
So, the roots of the auxiliary equation (2) are real and equal. Therefore, the general solution of equation (1) is
y = (A + Bx)e-x where A and B are two independent arbitrary constants ……….(3)
Given, y = 2 loge 2 when x = loge 2
Therefore, from (3) we get,
2 loge 2 = (A + B loge2)e-x
Or, 1/2(A + B loge2) = 2 log e2
Or, A + B loge2 = 4 loge2 ……….(4)
Again y = (4/3) loge3 when x = loge3
So, from (3) we get,
4/3 loge3 = (A + Bloge3)
Or, A + Bloge3 = 4loge3 ……….(5)
Now, (5) – (4) gives,
B(loge3 – loge2) = 4(loge3 – loge2)
⇒ B = 4
Putting B = 4 in (4) we get, A = 0
Thus the required solution of (1) is y = 4xe-x
So, C = 4
Or, (D2 + 2D+ 1)y = 0
⇒ d2y/dx2 + 2dy/dx + y = 0 ……….(1)
Let y = emx be a trial solution of equation (1). Then,
⇒ dy/dx = memx and d2y/dx2 = m2emx
Clearly, y = emx will satisfy equation (1). Hence, we have
⇒ m2.emx + 2m.emx + emx = 0
Or, m2 + 2m + 1 = 0 (as, emx ≠ 0) ………..(2)
Or, (m + 1)2 = 0
⇒ m = -1, -1
So, the roots of the auxiliary equation (2) are real and equal. Therefore, the general solution of equation (1) is
y = (A + Bx)e-x where A and B are two independent arbitrary constants ……….(3)
Given, y = 2 loge 2 when x = loge 2
Therefore, from (3) we get,
2 loge 2 = (A + B loge2)e-x
Or, 1/2(A + B loge2) = 2 log e2
Or, A + B loge2 = 4 loge2 ……….(4)
Again y = (4/3) loge3 when x = loge3
So, from (3) we get,
4/3 loge3 = (A + Bloge3)
Or, A + Bloge3 = 4loge3 ……….(5)
Now, (5) – (4) gives,
B(loge3 – loge2) = 4(loge3 – loge2)
⇒ B = 4
Putting B = 4 in (4) we get, A = 0
Thus the required solution of (1) is y = 4xe-x
So, C = 4
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What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge2 when x = loge2 and y = (4/3) loge3 when x = loge3?a)2b)-2c)-4d)4Correct answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge2 when x = loge2 and y = (4/3) loge3 when x = loge3?a)2b)-2c)-4d)4Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge2 when x = loge2 and y = (4/3) loge3 when x = loge3?a)2b)-2c)-4d)4Correct answer is option 'C'. Can you explain this answer?.
What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge2 when x = loge2 and y = (4/3) loge3 when x = loge3?a)2b)-2c)-4d)4Correct answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge2 when x = loge2 and y = (4/3) loge3 when x = loge3?a)2b)-2c)-4d)4Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for What will be the value of C if C the constant of the coefficient of the solution of the given equation (D + 1)2y = 0 given y = 2 loge2 when x = loge2 and y = (4/3) loge3 when x = loge3?a)2b)-2c)-4d)4Correct answer is option 'C'. Can you explain this answer?.
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