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Step 1The given IVP is as follows.Y′+3Y=6e−3tcos6t+24Y(0)=0Apply Laplace transform on both sides of the given differential equation as follows.LY′+3Y=L6e−3tcos6t+24LY′+3Y=L6e−3tcos6t+L24sLY−Y(0)+3LY=6Le−3tcos6t+L24(s+3)LY=6s+3(s+3)2+62+24sLY=6(s+3)2+62+24s(s+3)LY=6(s+3)2+62+8s−8(s+3)Step 2Apply inverse Laplace transform on both sides as follows.L−1{L{Y}}=L−1{6(s+3)2+62+8s−8(s+3)}Y(t)=L−1{6(s+3)2+62}+8L−1{1s}−8L−1{1(s+3)}Y(t)=e−3tsin6t+8−8L−1{1(s+3)}
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