I do not understand what "bounded" means. Anyways, let me solve the differencial equation maybe that would help.Originally Posted bybraddy

I think LaPlace Transforms is a primitive way to solve this.

Just use the charachteristic equation.

You have,

In such a case, there is a theorem. "The solution to a non-homogenenous equation is the sum of thegeneralsolution to the homogenous equation and aspecificsolution to the non-homogenous solution."

First solve the homogenous equation:

thus, its charachteristic is,

thus, . Thus, the general solutions to this homogenous equation are,

Second, find a specific solution to the non-homogenous,

. Since, there is a sine on the right look for solutions of the form,

thus,

thus,

.

Substitute that, into the non-homogenous

Thus,

Thus,

Solving, we have

Now, by the theorem, all general solution to this non-homogenous equation are the sum of these two solutions. Which gives,

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Looking at the initial conditions: Evaluating this at zero we getD,

Its, derivative is when evaluated at zero is,

Thus, solving these two equations,

Thus, the unique functions that satisfies this differencial equation with its initial conditions are:

Hope this helps.