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2dx/dt+dy/dt=5et & dy/dt-3dx/dt=5. y=0,x=0,t=0 |
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Answer» 2\(\frac{dx}{dt}\) + \(\frac{dy}{dt}\) = 5et .....(1) \(\frac{dy}{dt}\) - 3\(\frac{dx}{dt}\) = 5 .....(2) By subtracting equation (2) from (1), we get 5\(\frac{dx}{dt}\) = (et - 1) \(\frac{dx}{dt}\) = et-1 dx = (et - 1)dt x = et- t + c (By integrating both sides) at t = 0, x = 0 ∴ 0 = -e0 - 0 + c c = -e0 = -1 ∴ x = et - t -1 By putting x = et - t -1 in equation (2), we get \(\frac{dy}{dt}\) = 5 + 3\(\frac{d}{dt}\) (et- t -1) = 5 + 3et - 3 = 2 + 3et dy = (2+3et)dt y = 2t + 3et + c (By integrating both sides)given that at t = 0, y = 0 ∴ 0 = 2 x 0 + 3e0 + c c = -3e0 = -3 ∴y = 2t + 3et - 3 Hence, solution of given system of differential equation is x = et - t - 1 and y = 2t + 3et - 3 |
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