Anyone know the answer to these?
1 answer:
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Assuming you start with the homogeneous ODE,
upon taking the Laplace transform of both sides, you end up with
since the transform operator is linear, and the transform of 0 is 0.
I'll denote the Laplace transform of a function
into the 
-domain by 
.
Given the derivative of, its Laplace transform can be found easily from the definition of the transform itself:
Integrate by parts, setting
so that
The second term is just the transform of the original function, while the first term reduces to
since 
as 
, and 
as 
. So we have a rule for transforming the first derivative, and by the same process we can generalize it to any order provided that we're given the value of all the preceeding derivatives at 
.
The general rule gives us
and so our ODE becomes
Depending on how you learned about finding inverse transforms, you should either be comfortable with cross-referencing a table and do some pattern-matching, or be able to set up and compute an appropriate contour integral. The former approach seems to be more common, so I'll stick to that.
Recall that
and that given a function with transform 
, the shifted transform 
corresponds to the function 
.
We have
and so the inverse transform for our ODE is
and in case you're not familiar with hyperbolic functions, you have
upon taking the Laplace transform of both sides, you end up with
since the transform operator is linear, and the transform of 0 is 0.
I'll denote the Laplace transform of a function
Given the derivative of
Integrate by parts, setting
so that
The second term is just the transform of the original function, while the first term reduces to
The general rule gives us
and so our ODE becomes
Depending on how you learned about finding inverse transforms, you should either be comfortable with cross-referencing a table and do some pattern-matching, or be able to set up and compute an appropriate contour integral. The former approach seems to be more common, so I'll stick to that.
Recall that
and that given a function
We have
and so the inverse transform for our ODE is
and in case you're not familiar with hyperbolic functions, you have