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I won´t give the complete solutions here because in my opinion it is damaging for the purpose of learning to do everything by myself

note concerning the second equation posed :

the laplace transfomation ( LT ) of sen(x) = sin(t) I should assume to be known but I will give it here

LT[sin(t)] = 1/(1+s*s) ...s*s = s^2 etc..

the LT of y´´ = s*s*f(s)-s*f(0)-f´(0)
where f(s) = LT(the function sought)

the LT of y´= s*f(s) - f(0)

the LT of y = f(s)

thus - y´´+4*y´+ 8y = sen(t) can be

transformed easily with
LT(y´´ = s*s*f(s)-s and LT(y´=s*f(s)-1

where the boundary conditions have already been introduced

thus we get finally
-[s*s*f(s) -s] +4*[s*f(s)-1]+8*f(s)=1/(1+s*s)

this has fo be solved for f(s) and should be done by yourself

the resulting expression for f(s )will look something like
f(s) = 1[(1+s^2)*f2(s)] it
has to be expanded and should pose no difficulty in inverting giving the required solution

note concerning the second equation posed :

with the methods outlined above we only have to determine LT(t*y´

this is easily found to be
-d[f(s)]
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ds

the rest is completely simliar as before

hint for the first equation

after some simplification you will note from what has been given up until now that
an expression like ln[f(s)] should arise

In my opinion the ordinary method for solving is more convenient and besides that I have to stop doing subjects like this and go to work again

I do believe that now you are perfectly able to solve the first problem by yourself

thus the problems should have been reversed from the beginning in order to facilitate undestanding

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