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).
If the Particular Integral (PI) of the Differential equation(D2+a2)y= ...