Using the Dirac Delta function in a "Piecewise" ODE

Question

I have an equation:

$$y^{(4)}(x)-C_1\ u(x)y''(x) = C_2\ \delta(x)$$

where u(x) is the unit step function (centered at 0), δ(x) is the Dirac Delta function, and C1 and C2 are constants

I'm looking to try and solve this differential equation, and my first instinct is to turn it into a piecewise equation, but I'm not sure what to do about the Dirac Delta function. What do I do?

Answer 1

The singularity is at zero, so you just break it into two parts:

• On $x < 0$, it simplifies to $ y^{(4)}(x) = 0$
• On $x > 0$, it simplifies to $y^{(4)}(x) - C_1 y''(x) = 0$

And then once you've obtained the complete solution on each of the two regions, you then look for which pairs of functions can be assembled into something that satisfies the required condition near $x=0$.

Answer 2

As it stands, my answer is the following:

$$y^{(4)}(x) = 0,\ \ x < 0$$ $$y^{(4)}(x)-C_1y''(x) = C_2\ \delta(x), \ \ x \geq 0$$

The only thing to be careful with here is the bounds. If it were $x\leq0$, the equations will be different. I'm just hoping that the answer isn't somehow different.