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q2 - Key 1 Problem t 3 y 4 t 2 y = e-t I.C y-1 = 0,t<...

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MA366 Sathaye Quiz 2 Name/Sec. Consider the equation: t 3 y 0 + 4 t 2 y = e - t , I.C. y ( - 1) = 0 , t < 0 Solve the equation completely and show all work. Be sure to show these details: Equation rewritten in the standard form of a linear first order equation. The quantities p, q identified. The formula for the multiplier μ and its calculation. Integration of the equation. Use of the initial condition and the final answer.
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Unformatted text preview: Key 1. Problem: t 3 y + 4 t 2 y = e-t , I.C. y (-1) = 0 ,t < Rearrange: y + 4 t y = e-t t 3 , so p = 4 t ,q = e-t t 3 . 2. μ = exp( Z 4 t dt ) = exp(4 log( t )) = t 4 . 3. Multiplied equation: D t ( t 4 y ) = te-t so t 4 y = e-t (-t-1) + c 4. Using the initial condition: (-1) 4 (0) = e (1-1) + c so c = 0 . 5. Final answer: y = e-t (-t-1) t 4 . 2...
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