Solving two differential equations

[JM00404]

Please see the PDF attatchment to view the problems. Thank you for your time.

 

[arildno]

For 1:
What diff.eq does $$\psi(x)e^{-a_{1}\frac{x}{n}}$$ fulfill?

 

[saltydog]

Please see the PDF attatchment to view the problems. Thank you for your time.

The first one (too late I know but anyway):

Let's do a simple one first:

$$y^{''}+a_1y^{'}+a_2y=0$$

or:

$$(D^2+a_1D+a_2)y=0$$

or:

$$f(D)y=0$$

Now, use the exponential shift:

$$e^{cx}f(D)y=f(D-c)[e^{cx}y]$$

so up there, multiply by:

$$e^{a_1x/2}$$

so:

$$e^{a_1x/2}f(D)y=f(D-a_1/2)[e^{a_1x/2}\phi]=0$$

can you finish it?

$$f(D-a_1/2)=(D-a_1/2)^2+a_1(D-a_1/2)+a_2$$