method of undetermined coefficients 4y''+8y'+13y=1, y(0)=0, y'(0)=1/13

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method of undetermined coefficients 4y''+8y'+13y=1, y(0)=0, y'(0)=1/13

Mathematics
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use the quadratic formula find the roots to the characteristic polynomial \[4s ^{2}+8s+13=0\] then find all the solns to the corresponding homogenous D E . find a particular solution to the DE and put them together
i got to the point \[\left( -8\pm \sqrt{-144} \right)\div8\] bit confused with the negative square root
do i have to get imaginary numbers involved?

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Other answers:

\[\sqrt{-144}= 12i\]
\[y _{p}=1/13\] is a particular soln to the DE
the homo soln will be \[y _{h}(t)= c _{1}e ^{t}\cos8/12t+c _{2}e ^{t} \sin8/12\]
add the part. soln to this, then use your init values to solve for \[c _{1}, c _{2}\]
when u put 8/12 do you mean 12/8?
yes sorry
and can you put it in brackets please?
actually, i get that now, dont worry bout brackets
what do you mean
ok
thanks for all the help, i have the final answer as \[y(x)=\exp(x)[(2/39)\sin ((3/2)x)]+(1/13)\] do you know if that is right?

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