Need some assitance with this one. If you could give me some help I appreciate i

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Need some assitance with this one. If you could give me some help I appreciate it.

A solution to a differential equation is said to behave chaotically if any "small" change in initial conditions will eventually load to an exponential separation. That is, if the distance of initial conditions Delta (0) = for any epsilon (nonzero), Delta (t) grows exponentially. Consider a first order differential equation y' + p(t)y q(t) with initial condition y(0) = What conditions on p(t) and q(t) would make the solution behave chaotically? Consider a second order differential equation with constant coefficients ay" + by' + cy = f(t) with initial conditions y(0) = y0 and y'(0) = v0. What conditions on a, b, c, and f(t) would make the solution behave? Chaotically? Here, distance between solutions will be calculated as Delta (t) = (f(t) - g(t))2 + (f'(t) - g'(t))2. Consider the first: order linear system in two dimensions X' + AX = v with initial condition X(0) = X0. What conditions on the matrix A and the vector v would make the solution behave chaotically? Here, the distance between two solutions F(t) - F(t) and G(t) is given as ||F(t) - G(t)||2.

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