(Solved): (25 points) Consider the following integral equation, so called because the u ...

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(25 points) Consider the following integral equation, so called because the unknown dependent variable, \( y \), appears within an integral: \[ \int_{0}^{t} \sin (2(t-w)) y(w) d w=8 t^{2} \] This equation is defined for \( t \geq 0 \). a. Use convolution and Laplace transforms to find the Laplace transform of the solution. \[ Y(s)=\mathcal{L}\{y(t)\}= \] b. Obtain the solution \( y(t) \). \[ y(t)= \]