Verified answer

the first problem

y''+4y=4

L(y'')=s^2*Y(s)-s*y(0)-y'(0)

L(4y)=4*Y(s)

L(4)=1/s

then s^2*Y(s)-s*y(0)-y'(0)+4*Y(s)=1/s and y(0)=2,,, y'(0)=0

then s^2*Y(s)-2s+4*Y(s)=1/s

then Y(s)*(s^2+4)=(1/s)+2s

then Y(s)= ((1/s)+2s)/(s^2+4)

solve this equation using partial fraction or using Convolution theorem

then find laplace inverse of the function you will get y(t)

then differentiate y(t) to get y'(t) because we want to integrate it using limited integrate from 0 to 2 pi

so integral of y'(t) = integral from t=0 to t=2pi ( of the resultant)

so definite integration you will able to find y(t) in terms of t

Hope that I helped you :)

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