If y=Asinx+Bcosx, then prove that dx2d2y+y=0.
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Step-by-Step Solution
Step 1: Differentiate the given equation with respect to x
We start by finding the first derivative of the given function y with respect to x. We apply the basic differentiation rules: the derivative of sinx is cosx, and the derivative of cosx is −sinx. The constants A and B remain as coefficients.
Step 2: Differentiate the first derivative again with respect to x
Next, we find the second derivative by differentiating the first derivative, dxdy, with respect to x. Again, we apply the differentiation rules: the derivative of cosx is −sinx, and the derivative of sinx is cosx.
Step 3: Substitute the second derivative into the expression
Now we substitute the expression for dx2d2y that we just found, and the original expression for y, into the equation we need to prove: dx2d2y+y=0.
Step 4: Simplify the expression
By combining like terms, we can see that the terms −Asinx and Asinx cancel each other out, and similarly, −Bcosx and Bcosx cancel each other out. This results in a sum of zero, thus proving the given statement.