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If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2x is of the form eαx [(α'x2 + α") sin βx + β' x cos βx] then find the value of α".
Correct answer is '3'. Can you explain this answer?
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If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis o...
The correct answer is: 3
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If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis o...
To find the particular integral (PI) of the given differential equation, we can assume that the PI has the form of the given function, i.e., e^ax^2e^2xsin(2x).
Differentiating the assumed PI twice:
First derivative: (2ae^ax^2e^2xsin(2x) + e^ax^2e^2x(2sin(2x) + 4xcos(2x)))
Second derivative: (4ae^ax^2e^2xsin(2x) + 4e^ax^2e^2x(2sin(2x) + 4xcos(2x)) + e^ax^2e^2x(-4sin(2x) + 8xcos(2x)))
Now we substitute these derivatives into the differential equation:
(4ae^ax^2e^2xsin(2x) + 4e^ax^2e^2x(2sin(2x) + 4xcos(2x)) + e^ax^2e^2x(-4sin(2x) + 8xcos(2x)) - 4(2ae^ax^2e^2xsin(2x) + e^ax^2e^2x(2sin(2x) + 4xcos(2x))) + 4e^ax^2e^2xsin(2x)) = 8x^2e^2xsin(2x)
Simplifying and canceling out terms:
4e^ax^2e^2x(2sin(2x) + 4xcos(2x) - 4sin(2x) + 8xcos(2x)) = 8x^2e^2xsin(2x)
8e^ax^2e^2x(2xcos(2x) + 2xcos(2x)) = 8x^2e^2xsin(2x)
16x^2e^4xsin(2x) = 8x^2e^2xsin(2x)
e^4x = e^2x
Since the exponential terms are equal, we can equate the exponents:
4x = 2x
2x = 0
x = 0
Therefore, the PI of the given differential equation is e^0 = 1.
Differentiating the assumed PI twice:
First derivative: (2ae^ax^2e^2xsin(2x) + e^ax^2e^2x(2sin(2x) + 4xcos(2x)))
Second derivative: (4ae^ax^2e^2xsin(2x) + 4e^ax^2e^2x(2sin(2x) + 4xcos(2x)) + e^ax^2e^2x(-4sin(2x) + 8xcos(2x)))
Now we substitute these derivatives into the differential equation:
(4ae^ax^2e^2xsin(2x) + 4e^ax^2e^2x(2sin(2x) + 4xcos(2x)) + e^ax^2e^2x(-4sin(2x) + 8xcos(2x)) - 4(2ae^ax^2e^2xsin(2x) + e^ax^2e^2x(2sin(2x) + 4xcos(2x))) + 4e^ax^2e^2xsin(2x)) = 8x^2e^2xsin(2x)
Simplifying and canceling out terms:
4e^ax^2e^2x(2sin(2x) + 4xcos(2x) - 4sin(2x) + 8xcos(2x)) = 8x^2e^2xsin(2x)
8e^ax^2e^2x(2xcos(2x) + 2xcos(2x)) = 8x^2e^2xsin(2x)
16x^2e^4xsin(2x) = 8x^2e^2xsin(2x)
e^4x = e^2x
Since the exponential terms are equal, we can equate the exponents:
4x = 2x
2x = 0
x = 0
Therefore, the PI of the given differential equation is e^0 = 1.
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If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis of the form eαx [(α'x2 +α") sinβx +β' x cosβx]then find the value ofα".Correct answer is '3'. Can you explain this answer? for Physics 2025 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis of the form eαx [(α'x2 +α") sinβx +β' x cosβx]then find the value ofα".Correct answer is '3'. Can you explain this answer? covers all topics & solutions for Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis of the form eαx [(α'x2 +α") sinβx +β' x cosβx]then find the value ofα".Correct answer is '3'. Can you explain this answer?.
If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis of the form eαx [(α'x2 +α") sinβx +β' x cosβx]then find the value ofα".Correct answer is '3'. Can you explain this answer? for Physics 2025 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis of the form eαx [(α'x2 +α") sinβx +β' x cosβx]then find the value ofα".Correct answer is '3'. Can you explain this answer? covers all topics & solutions for Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the PI of differential equation (D2 - 4D + 4) y = 8x2e2x sin 2xis of the form eαx [(α'x2 +α") sinβx +β' x cosβx]then find the value ofα".Correct answer is '3'. Can you explain this answer?.
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