If cosx and sinx are solutions of the different equation a0d2ydx2+a1dydx+a2y=0, where a0,a1,a2 are real constants, then which of the following is/are always true?
A
Acosx+Bsinx is a solution, where A and B are real costants
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B
Acos(x+π4) is a solution, where A is real constant
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C
Acosxsinx is a solution, where A is real constant
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D
Acos(x+π4)+Bsin(x−π4) is a solution, where A and B are real constants
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Solution
The correct options are AAcosx+Bsinx is a solution, where A and B are real costants BAcos(x+π4) is a solution, where A is real constant DAcos(x+π4)+Bsin(x−π4) is a solution, where A and B are real constants Given differential equation is a0d2ydx2+a1dydx+a2y=0
Also given that, cosx and sinx are solutions to the given differential equation.
⟹Acosx+Bsinx is a solution to the given differential equation as it is homogenous, where A and B are real constants.
Acos(x+Π4)=A√2(−sinx+cosx)=C1sinx+C2cosx is also a solution where A=−B