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

The year before the 10s so it was like when caveman were alive