Mathematics

Mathematics

Problem

Suppose differentiable functions x(t)x(t) and y(t)y(t) satisfy dxdt=5y,dydt=−5x\frac{dx}{dt}=5y, \qquad \frac{dy}{dt}=-5x for all tt. Which of the following second-order differential equations involving only xx must be true? a) d2xdt2=−x\displaystyle\frac{d^{2}x}{dt^{2}}=-x b) d2xdt2=−25x\displaystyle\frac{d^{2}x}{dt^{2}}=-25x c) d2xdt2=25x\displaystyle\frac{d^{2}x}{dt^{2}}=25x d) d2xdt2=−5x\displaystyle\frac{d^{2}x}{dt^{2}}=-5x

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