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Find the Laplace transform of the following differential equation and express it \$$Q(s)\$$ as a single fraction:

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\$$\\frac{d^2q}{dt^2}+\\var{a}\\frac{dq}{dt}+\\var{b}q(t)=\\var{c}e^{-\\var{d}t}\$$    where   \$$q(0)=\\var{f}\$$ and  \$$q'(0)=\\var{g}\$$

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\$$Q(s)=\$$ [[0]]

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\$$\\frac{d^2q}{dt^2}+\\var{a}\\frac{dq}{dt}+\\var{b}q(t)=\\var{c}e^{-\\var{d}t}\$$    where   \$$q(0)=\\var{f}\$$ and  \$$q'(0)=\\var{g}\$$

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\$$s^2Q(s)-sq(0)-q'(0)+\\var{a}(s(Q(s)-q(0))+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}\$$

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\$$s^2Q(s)-\\var{f}s-\\var{g}+\\var{a}sQ(s)-\\var{a}*\\var{f}+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}\$$

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\$$s^2Q(s)+\\var{a}sQ(s)+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}+\\var{f}s+\\simplify{{g}+{a}*{f}}\$$

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\$$(s^2+\\var{a}s+\\var{b})Q(s)=\\frac{\\var{c}+(\\var{f}s+\\simplify{{g}+{a}*{f}})(s+\\var{d})}{s+\\var{d}}\$$

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\$$Q(s)=\\frac{\\simplify{{f}s^2+({a}*{f}+{g}+{d}*{f})s+(({g}+{f}*{a})*{d}+{c})}}{(s+\\var{d})(s^2+\\var{a}s+\\var{b})}\$$

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