Determine if the function *f* (*x*) = sin *x* is a solution to the IVP

*f* ^{(3)}(*x*) = -*f* '(*x*) and *f* (0) = 0.

Answer

The initial condition is satisfied, since sin (0) = 0. We need some derivatives to check the differential equation:

*f*'(*x*) = cos *x*

*f* ^{(2)}(*x*) = -sin *x*

*f* ^{(3)}(*x*) = -cos *x*.

The left-hand side of the d.e. is

*f* ^{(3)}(*x*) = -cos *x*.

The right-hand side of the d.e. is

-*f* '(*x*) = -cos *x*.

Since the two sides of the d.e. agree, the function *f* (*x*) = sin *x* is a solution to the differential equation. Since the function *f* (*x*) = sin *x* satisfies both the d.e. and the initial condition, this function is a solution to the IVP.