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Math Help - Verify that the given function is a solution of the differential equations

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    Verify that the given function is a solution of the differential equations

    Verify that the given function is a solution of the DE and A and B are constants.

    (a) y + y = 0; y = A cos x + B sin x

    How do I start this problem??? I have no clue where to start. Thank you.
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  2. #2
    fkf
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    Re: Verify that the given function is a solution of the differential equations

    Quote Originally Posted by tjsdndnjs View Post
    Verify that the given function is a solution of the DE and A and B are constants.

    (a) y + y = 0; y = A cos x + B sin x

    How do I start this problem??? I have no clue where to start. Thank you.
    We have
    y''+y=0

    And a statement that says that y is a solution where
    y = Acos(x)+Bsin(x)

    Which implies that
    y = Acos(x)+Bsin(x) => y' = -Asin(x)+Bcos(x) => y'' = -Acos(x)-Bsin(x)

    If we go back to y''+y'=0 we can check if the left side equals 0 when using the given derivatives of the function above. Hence
    y''+y = -Acos(x)-Bsin(x)+Acos(x)+Bsin(x) = 0

    Which solves the DE and hence Q.E.D
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    Re: Verify that the given function is a solution of the differential equations

    Alternatively, u can check that y_1 = \cos x & y_2 = \sin x are 2 solutions for the homogenous ODE. Therefore a linear combination of y_1 and y_2 is also a solution to this homogenous ODE, i.e. y = A\cos x + B\sin x also satisfies the given ODE.
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