# Math Help - How to solve d^2y/dx^2 = k^2y

1. ## How to solve d^2y/dx^2 = k^2y

How do you solve

d^2y/dx^2 = k^2 y

for boundary conditions:

y = y0 when x =0
y = 0 when x = infinity

Any help would be much appreciated.

2. ## Re: How to solve d^2y/dx^2 = k^2y

Originally Posted by dbbwgr54
How do you solve

d^2y/dx^2 = k^2 y

for boundary conditions:

y = y0 when x =0
y = 0 when x = infinity

Any help would be much appreciated.
Dear dbbwgr54,

Use the trial solution $y = Ae^{mx}$.

$\Rightarrow\frac{dy}{dx}=Ame^{mx}$

$\Rightarrow\frac{d^{2}y}{dx^2}=Am^{2}e^{mx}$

Therefore the auxiliary equation will be,

$m^2-k^2=0$

$\Rightarrow m = \pm k$

Hope you can continue.

3. ## Re: How to solve d^2y/dx^2 = k^2y

Thank you Sudharaka, you help is much appreciated.

4. ## Re: How to solve d^2y/dx^2 = k^2y

Originally Posted by dbbwgr54
Thank you Sudharaka, you help is much appreciated.

Not quite. You actually have to include both solutions, at first, and then use your boundary conditions to determine the two arbitrary constants. Remember: the solution to a second-order DE should always two arbitrary constants. So what will your general solution, with two arbitrary constants, look like?

5. ## Re: How to solve d^2y/dx^2 = k^2y

I think I'm lost. Sorry for this but I do not understand how the auxiliary equation gives m^2 - k^2.

Also, is this treated as a homogeneous or in-homogeneous equation?

Thanks for the help!

6. ## Re: How to solve d^2y/dx^2 = k^2y

Originally Posted by dbbwgr54
How do you solve

d^2y/dx^2 = k^2 y

for boundary conditions:

y = y0 when x =0
y = 0 when x = infinity

Any help would be much appreciated.
The DE can be written as $\frac{d^2 y}{dx^2} - k^2 y = 0$. It is obviously homogeneous.

Now refer to your class notes and textbook on how to solve second order DE's with constant coefficients.

Alternatively, you can substitute $\frac{d^2 y}{dx^2} = \frac{d \left( \frac{v^2}{2}\right)}{dy}$ where $v = \frac{dy}{dx}$ and then integrate directly with respect to y and then solve the resulting 1st order DE.

As always, the techniques, applications and examples are found in the class notes and textbook.