Solve the initial value problem

[jegues]

Homework Statement

Solve the following initial value problem

$$y^{''} + 2y^{'} - 15y = 4\delta(t-2), \quad y(0) = 1, \quad y'(0) = -1.$$

Homework Equations

The Attempt at a Solution

See figure attached for my attempt at the solution.

Does anyone see any problems? Sorry if it's kinda crunched in there.

 

[HallsofIvy]

The characteristic equation for this d.e. is $r^2+ 2r- 15= 0$ which has real roots r= -5 and r= 3. The solution cannot possibly involve "sine" or "cosine". I suspect you have a sign error.

 

[jegues]

The characteristic equation for this d.e. is $r^2+ 2r- 15= 0$ which has real roots r= -5 and r= 3. The solution cannot possibly involve "sine" or "cosine". I suspect you have a sign error.

Hmmm... I can't seem to spot any errors. We haven't learned anything about the characteristic equations so I can't really relate to what you're telling me.

Is this a problem that should be solved with another method instead of using laplace transforms?

 

[Saladsamurai]

Hi there jegues!

What is the inverse Laplace of 1 / (s + a) ?

 

[HallsofIvy]

I have always disliked Laplace transform- quite frankly there is NO problem that can be solve by Laplace transform that cannot be solved by some other method (and probably simpler). What I would do with this problem is to first get the two independent solutions to the associated homogeneous equation, y''+ 2y'- 15y= 0 by solving the characteristic equation $r^2+ 2r- 15= 0$, then using "variation of parameters" to solve the entire equation.

As Saladsamurai suggests, you appear to be confusing the inverse Laplace transform of $1/(s+ a)$ with that of $1/(s^2+ a^2)$.

 

[jegues]

I have always disliked Laplace transform- quite frankly there is NO problem that can be solve by Laplace transform that cannot be solved by some other method (and probably simpler). What I would do with this problem is to first get the two independent solutions to the associated homogeneous equation, y''+ 2y'- 15y= 0 by solving the characteristic equation $r^2+ 2r- 15= 0$, then using "variation of parameters" to solve the entire equation.

As Saladsamurai suggests, you appear to be confusing the inverse Laplace transform of $1/(s+ a)$ with that of $1/(s^2+ a^2)$.

Whoops there's my mistake!

This clears things up now.

Thanks again!

EDIT: I also made a mistake in my partial fractions decomposition.