// Numbas version: exam_results_page_options {"name": "Differential equation with 3 simple linear factors: Algebraic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

Solve the differential equation:

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\$$\\frac{d^2i}{dt^2}+\\simplify{{a1}+{b1}}\\frac{di}{dt}+\\simplify{{a1}*{b1}}i(t)=\\var{c1}e^{-\\var{d1}t}\$$  where \$$i(0)=\\var{i0} \\,\\, and \\,\\, i'(0)=\\var{i1}\$$

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Solve a Differential equation with 3 simple linear factors

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\$$i(t)=\$$ [[0]]

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\$$\\frac{d^2i}{dt^2}+\\simplify{{a1}+{b1}}\\frac{di}{dt}+\\simplify{{a1}*{b1}}i(t)=\\var{c1}e^{-\\var{d1}t}\$$  where \$$i(0)=\\var{i0} \\,\\, and \\,\\, i'(0)=\\var{i1}\$$

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The Laplace transform of this is given by:

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\$$s^2I(s)-\\var{i0}s-\\var{i1}+\\simplify{{a1}+{b1}}(sI(s)-\\var{i0})+\\simplify{{a1}*{b1}}I(s)=\\frac{\\var{c1}}{s+\\var{d1}}\$$

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Gathering only \$$I(s)\$$ terms on the left hand side and factoring gives:

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\$$(s^2+\\simplify{{a1}+{b1}}s+\\simplify{{a1}*{b1}})I(s)=\\frac{\\var{c1}}{s+\\var{d1}}+\\var{i0}s+\\simplify{{i1}+({a1}+{b1})*{i0}}\$$

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\$$(s^2+\\simplify{{a1}+{b1}}s+\\simplify{{a1}*{b1}})I(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{b1})*{i0})*(s+{d1})}}{s+\\var{d1}}\$$

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\$$I(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{b1})*{i0})*(s+{d1})}}{(s+\\var{d1})(s+\\var{a1})(s+\\var{b1})}\$$

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\$$I(s)=\\frac{A}{s+\\var{d1}}+\\frac{B}{s+\\var{a1}}+\\frac{C}{s+\\var{b1}}\$$

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\$$\\simplify{{c1}+({i0}s+{i1}+({a1}+{b1})*{i0})*(s+{d1})}=A(s+\\var{a1})(s+\\var{b1})+B(s+\\var{d1})(s+\\var{b1})+C(s+\\var{d1})(s+\\var{a1})\$$

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Let \$$s=-\\var{d1}\$$

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\$$\\simplify{{c1}+({i0}*-{d1}+{i1}+({a1}+{b1})*{i0})*(-{d1}+{d1})}=\\simplify{(-{d1}+{a1})(-{d1}+{b1})}A\$$

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\$$A=\\simplify{({c1}+({i0}*-{d1}+{i1}+({a1}+{b1})*{i0})*(-{d1}+{d1}))/((-{d1}+{a1})(-{d1}+{b1}))}\$$

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Let \$$s=-\\var{a1}\$$

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\$$\\simplify{{c1}+({i0}*-{a1}+{i1}+({a1}+{b1})*{i0})*(-{a1}+{d1})}=\\simplify{(-{a1}+{d1})(-{a1}+{b1})}B\$$

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\$$B=\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{b1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1})(-{a1}+{b1}))}\$$

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Let \$$s=-\\var{b1}\$$

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\$$\\simplify{{c1}+({i0}*-{b1}+{i1}+({a1}+{b1})*{i0})*(-{b1}+{d1})}=\\simplify{(-{b1}+{d1})(-{b1}+{a1})}C\$$

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\$$C=\\simplify{({c1}+({i0}*-{b1}+{i1}+({a1}+{b1})*{i0})*(-{b1}+{d1}))/((-{b1}+{d1})(-{b1}+{a1}))}\$$

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\$$i(t)=\\simplify{({c1}+({i0}*-{d1}+{i1}+({a1}+{b1})*{i0})*(-{d1}+{d1}))/((-{d1}+{a1})(-{d1}+{b1}))}e^{-\\var{d1}t}+\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{b1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1})(-{a1}+{b1}))}e^{-\\var{a1}t}+\\simplify{({c1}+({i0}*-{b1}+{i1}+({a1}+{b1})*{i0})*(-{b1}+{d1}))/((-{b1}+{d1})(-{b1}+{a1}))}e^{-\\var{b1}t}\$$

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