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## Homework Statement

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Its been a pretty long time since I've taken differential equations and I'm expected to know the solutions to the kinds of DEs below for my fluid mechanics class.

In class my professor worked a 2nd order DE: dy

^{2}/dx

^{2}= -k

^{2}*y

and told us the way to think about it was to ask yourself "what function gives an alternating solution when derived?" So then the solution he gave us was y = C

_{1}*sin(kx)+C

_{2}*cos(kx).

I understand that when you integrate/derive a sin or cos function it gives back a sin or cos, but I don't understand how you would know that the function needed to repeat itself... I feel like I didn't really get an explanation as to how you would even know to ask that question to help you memorize the solution. I hope that makes sense.

Another 2nd order DE he worked was: dy

^{2}/dx

^{2}= k

^{2}*y

and told us the way to think about it was to ask yourself "what function when differentiated twice gives back the same function?" And the solution was:

y = C

_{1}*e

^{kx}+C

_{2}*e

^{-kx}. And again, I understand that e

^{x}derived/ integrated is e

^{x}, but how does thinking this help me memorize the solution?

I also didn't understand this example: dv/dt = -k*v

Solution: v = C

_{1}*e

^{-kt}

Thanks for any help!