// Numbas version: exam_results_page_options {"name": "Laplace: Inverse Laplace Completing the Square", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "name": "Laplace: Inverse Laplace Completing the Square", "parts": [{"sortAnswers": false, "prompt": "

Write down the inverse Laplace transform

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\$$\\mathscr{L}^{-1}\\{F(s)\\}=\$$ [[0]]

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\$$Q(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\$$

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\$$Q(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}s+\\var{C}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}}\$$

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\$$Q(s)=\\frac{\\var{A}}{s+\\var{a1}}+\\frac{\\var{B}(s+\\var{b1})-\\simplify{{B}*{b1}-{C}}}{(s+\\var{b1})^2+\\simplify{{c1}-{b1}^2}}\$$

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\$$q(t)=\\var{A}e^{\\var{a1}t}+\\var{B}e^{-\\var{b1}t}cos\\left(\\sqrt{\\simplify{{c1}-{b1}^2}}t\\right)+\\frac{-\\simplify{{B}*{b1}-{C}}}{\\sqrt{\\simplify{{c1}-{b1}^2}}}e^{-\\var{b1}t}sin\\left(\\sqrt{\\simplify{{c1}-{b1}^2}}t\\right)\$$

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Determine the inverse Laplace Transform of the following using completion of the square.

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\$$F(s)=\\dfrac{\\var{B}s+\\var{C}}{s^2+\\simplify{{b1}*2}s+\\var{c1}}\$$

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