Find the laplace transform of y''+7y'+7y given that y(0)=0 and y'(0)=7

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