You must be able to solve the following ODEs.
You should be able to immediately write down the general solution
in each case:
First order: exponential decay
ODE
\frac{\td Y}{\td t} = -\mu Y
Solution
Y (t) = A \ee ^ {-\mu t}
Second order: trigonometric
ODE
\frac{\td^2 Y}{{\td x}^2} = -\lambda^2 Y
Solution
Y (x) = A \cos (\lambda x) + B \sin (\lambda x)
Note that \cos is an even function and \sin is an odd function:
If Y is zero at x = 0, you only want \sin
If Y has zero slope at x = 0, you only want \cos.
Second order: hyperbolic (or exponential)
ODE
\frac{\td^2 Y}{{\td x}^2} = +\lambda^2 Y
Solution
Y (x) = A \cosh (\lambda x) + B \sinh (\lambda x)
OR
Y (x) = C \ee^ {\lambda x} + D \ee ^ {-\lambda x}
When solving boundary value problems,
the first form with the hyperbolic functions\cosh and \sinh
is much nicer to work with
because \cosh is an even function and \sinh is an odd function:
If Y is zero at x = 0, you only want \sinh
If Y has zero slope at x = 0, you only want \cosh.