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Enter the value for \$$A\$$ as an exact fraction.            \$$A=\$$ [[0]]

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Enter the value for \$$B\$$ as an exact fraction.            \$$B=\$$ [[1]]

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Enter the value for \$$C\$$ as an exact fraction.            \$$C=\$$ [[2]]

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"}], "statement": "

The diagram below shows a typical mass-spring-damper system as might apply to the suspension of a car.

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(Masses have mass M, springs with stiffness k and dampers having damping coefficient B).

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The associated variables are displacement x(t) and force F(t).

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Initially the mass is at a distance \$$\\var{b1}cm\$$ from the equilibrium point and is moving at \$$\\var{b2}cm/s\$$.

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If \$$B=\\simplify{2*{a1}}\$$ and  \$$k=\\simplify{{a1}^2+{a2}^2}\$$ and the system is subjected to an external applied force \$$F(t)=\\var{R}e^{-\\var{d1}t}\$$

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then from Newton's law we get the differential equation:

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\$$\\frac{d^2x}{dt^2}+\\simplify{2*{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}x(t)=\\var{R}e^{-\\var{d1}t}\$$   where \$$x(0)=\\var{b1} \\,\\, and \\,\\, x'(0)=\\var{b2}\$$

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The solution of the equation is given by:

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\$$x(t)=Ae^{-\\var{d1}t}+Be^{-\\var{a1}t}cos(\\var{a2}t)+Ce^{-\\var{a1}t}sin(\\var{a2}t)\$$

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.

", "rulesets": {}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "functions": {}, "advice": "

\$$\\frac{d^2x}{dt^2}+\\simplify{2*{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}x(t)=\\var{R}e^{-\\var{d1}t}\$$  where \$$x(0)=\\var{b1} \\,\\, and \\,\\, x'(0)=\\var{b2}\$$

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The Laplace transform of this is given by:

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\$$s^2X(s)-\\var{b1}s-\\var{b2}+\\simplify{2*{a1}}(sX(s)-\\var{b1})+\\simplify{{a1}*{a1}+{a2}*{a2}}X(s)=\\frac{\\var{R}}{s+\\var{d1}}\$$

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

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\$$s^2X(s)+\\simplify{2*{a1}}sX(s)+\\simplify{{a1}^2+{a2}^2}X(s)=\\frac{\\var{R}}{s+\\var{d1}}+\\var{b1}s+\\simplify{{b2}+2*{a1}*{b1}}\$$

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\$$(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})X(s)=\\frac{\\var{b1}s^2+\\simplify{{b1}*{d1}+{b2}+2*{a1}*{b1}}s+\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}}{s+\\var{d1}}\$$

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\$$X(s)=\\frac{\\var{b1}s^2+\\simplify{{b1}*{d1}+{b2}+2*{a1}*{b1}}s+\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}}{(s+\\var{d1})(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\$$

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\$$X(s)=\\frac{A}{s+\\var{d1}}+\\frac{Bs+c}{(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\$$

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\$$\\var{b1}s^2+\\simplify{{b1}*{d1}+{b2}+2*{a1}*{b1}}s+\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}=A(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})+Bs(s+\\var{d1})+c(s+\\var{d1})\$$

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

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\$$\\var{R}=A(\\simplify{{d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2})\$$

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\$$A=\\frac{\\var{R}}{(\\simplify{{d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2})}\$$

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Compare the coefficients of \$$s^2\$$

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\$$\\var{b1}=A+B\$$

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\$$B=\\var{b1}-\\frac{\\var{R}}{(\\simplify{{d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2})}=\\simplify{({b1}*({d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2)-{R})/({d1}^2-2*{a1}*{d1}+{a1}^2+{a2}^2)}\$$

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Let \$$s=0\$$

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\$$\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}=\\simplify{({a1}^2+{a2}^2)}A+c(\\var{d1})\$$

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\$$\\var{d1}c=\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}-\\simplify{({a1}^2+{a2}^2)}A\$$

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\$$\\var{d1}c=\\simplify{{b2}*{d1}+2*{a1}*{b1}*{d1}+{R}}-\\simplify{({a1}^2+{a2}^2)}\\left(\\frac{\\var{R}}{\\var{A_denom}}\\right)\$$

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\$$c=\\frac{\\var{C_numer}}{\\simplify{{d1}*{A_denom}}}=\\simplify{{c_numer}/({d1}*{A_denom})}\$$

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\$$X(s)=\\frac{\\simplify{{R}/{A_denom}}}{s+\\var{d1}}+\\frac{\\simplify{{B_numer}/{A_denom}}s +\\simplify{{c_numer}/({d1}*{A_denom})}}{ s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2}}\$$

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\$$X(s)=\\frac{\\simplify{{R}/{A_denom}}}{s+\\var{d1}}+\\frac{\\simplify{{B_numer}/{A_denom}}(s+\\var{a1}) +\\simplify{({c_numer}-{B_numer}*{a1}*{d1})/({d1}*{A_denom})}}{(s+\\var{a1})^2+\\simplify{{a2}^2}}\$$

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\$$X(s)=\\frac{\\simplify{{R}/{A_denom}}}{s+\\var{d1}}+\\frac{\\simplify{{B_numer}/{A_denom}}(s+\\var{a1})}{(s+\\var{a1})^2+\\simplify{{a2}^2}}+\\frac{\\simplify{({c_numer}-{B_numer}*{a1}*{d1})/({d1}*{A_denom})}}{(s+\\var{a1})^2+\\simplify{{a2}^2}}\$$

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\$$x(t)=\\simplify{{R}/{A_denom}}e^{-\\var{d1}t}+\\simplify{{B_numer}/{A_denom}}e^{-\\var{a1}t}cos(\\var{a2}t)+\\simplify{({c_numer}-{B_numer}*{a1}*{d1})/({d1}*{A_denom})}e^{-\\var{a1}t}\\frac{1}{\\var{a2}}sin(\\var{a2}t)\$$

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