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If the Particular Integral (PI) of the Differential equation (D2 + a2)y = cos ax is given by f(x). Then find the value of a2f (π/2a).
Correct answer is '0.785'. Can you explain this answer?
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If the Particular Integral (PI) of the Differential equation(D2+a2)y= ...
As, we known, the P.I. of (D2 + a2)y = cos ax is given as x/2a sin ax
∴
∴
Now, 
The correct answer is: 0.785
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If the Particular Integral (PI) of the Differential equation(D2+a2)y= ...
To find the particular integral of the differential equation, we can assume a particular solution of the form f(x) = A cos(ax) + B sin(ax), where A and B are constants to be determined.
Taking the second derivative of f(x) with respect to x, we have:
f''(x) = -A a^2 cos(ax) - B a^2 sin(ax)
Substituting this into the differential equation, we get:
(-A a^2 cos(ax) - B a^2 sin(ax)) + a^2 (A cos(ax) + B sin(ax)) = cos(ax)
Simplifying this equation, we have:
(-A a^2 + a^2 A) cos(ax) + (-B a^2 + a^2 B) sin(ax) = cos(ax)
Since cos(ax) and sin(ax) are linearly independent, the coefficients of cos(ax) and sin(ax) must be equal on both sides of the equation. Therefore,
-A a^2 + a^2 A = 1 (1)
-B a^2 + a^2 B = 0 (2)
From equation (2), we have -Ba^2 + a^2 B = 0, which implies B = 0.
Substituting B = 0 into equation (1), we have -A a^2 + a^2 A = 1, which simplifies to A = -1.
Therefore, the particular solution f(x) is given by:
f(x) = -cos(ax)
To find the value of a^2 f(x), we substitute f(x) = -cos(ax) into the expression:
a^2 f(x) = a^2 (-cos(ax)) = -a^2 cos(ax)
So, the value of a^2 f(x) is -a^2 cos(ax).
Taking the second derivative of f(x) with respect to x, we have:
f''(x) = -A a^2 cos(ax) - B a^2 sin(ax)
Substituting this into the differential equation, we get:
(-A a^2 cos(ax) - B a^2 sin(ax)) + a^2 (A cos(ax) + B sin(ax)) = cos(ax)
Simplifying this equation, we have:
(-A a^2 + a^2 A) cos(ax) + (-B a^2 + a^2 B) sin(ax) = cos(ax)
Since cos(ax) and sin(ax) are linearly independent, the coefficients of cos(ax) and sin(ax) must be equal on both sides of the equation. Therefore,
-A a^2 + a^2 A = 1 (1)
-B a^2 + a^2 B = 0 (2)
From equation (2), we have -Ba^2 + a^2 B = 0, which implies B = 0.
Substituting B = 0 into equation (1), we have -A a^2 + a^2 A = 1, which simplifies to A = -1.
Therefore, the particular solution f(x) is given by:
f(x) = -cos(ax)
To find the value of a^2 f(x), we substitute f(x) = -cos(ax) into the expression:
a^2 f(x) = a^2 (-cos(ax)) = -a^2 cos(ax)
So, the value of a^2 f(x) is -a^2 cos(ax).
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If the Particular Integral (PI) of the Differential equation(D2+a2)y= cosaxis given byf(x).Then find the value of a2f (π/2a).Correct answer is '0.785'. Can you explain this answer?
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If the Particular Integral (PI) of the Differential equation(D2+a2)y= cosaxis given byf(x).Then find the value of a2f (π/2a).Correct answer is '0.785'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about If the Particular Integral (PI) of the Differential equation(D2+a2)y= cosaxis given byf(x).Then find the value of a2f (π/2a).Correct answer is '0.785'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the Particular Integral (PI) of the Differential equation(D2+a2)y= cosaxis given byf(x).Then find the value of a2f (π/2a).Correct answer is '0.785'. Can you explain this answer?.
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