# Thread: Verify that the given function is a solution of the differential equations

1. ## 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.

2. ## Re: Verify that the given function is a solution of the differential equations

Originally Posted by tjsdndnjs
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

3. ## 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.