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Question

If cosx and sinx are solutions of the differential equation a0d2ydx2+a1dydx+a2y=0, where a0,a1 and a2 are all real constants, then which of the following is/are always true?

A
A cosx+Bsinx is a solution, where A and B real constants
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B
A cos(x+π4) is a solution, where A is real constant
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C
A cosxsinx is a solution, where A is real constant
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D
A cos(x+π4)+Bsin(xπ4) is a solution, where A and B are real constants
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Solution

The correct options are
A A cos(x+π4) is a solution, where A is real constant
B A cosx+Bsinx is a solution, where A and B real constants
D A cos(x+π4)+Bsin(xπ4) is a solution, where A and B are real constants
(a) Let f(x)=cosx and g(x)=sinx
Consider the Wronskian of f(x) and g(x),
W=f(x)g(x)f(x)g(x)

=cosxsinxsinxcosx

=cos2x+sin2x=10

Thus, the functions are linearly independent. So, the general solution of given differential equation is given by y=Acosx+Bsinx, where A and B are real constants.

[ if y1 and y2 are linearly independent solutions of the differential equation ay"+by+c=0, then the general solution is

y=c1y1+C2y2, where c1 and c2 are constants]
Hence, option (a) is true.

(b) Let y=Acos(x+π4)
=A(cosxcosπ4sinxsinπ4)
=A2(cosxsinx)
=A2cosx+(A2)sinx
which is in the form of general solution.
Hence, option (b) is true.

(c) Let y=Acosxsinx, Which cannot be expressed in the form of general solution.

(d) Let y=Acos(x+π4)+Bsin(xπ4)
=A(cosx12sinx12)+B(sinx12cosx12)
=cosx(A2B2)+sinx(B2A2) which is in the form of general solution.
Hence, option (d) is true.

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