Consider the following differential equation. (A computer algebra system is reco

Question

Consider the following differential equation. (A computer algebra system is recommended.) (1 + t^2)y' + 4 ty = (1 + t^2)^-2 Draw a direction field for the given differential equation. Based on an inspection of the direction field, describe how solutions behave for large t. The solutions appear to be oscillatory. All solutions seem to converge to the function y_0(t) = 4. All solutions seem to converge to the function y_0(t) = 0. All solutions seem to eventually have negative slopes, and hence decrease without bound. All solutions seem to eventually have positive slopes, and hence increase without bound. Find the general solution of the given differential equation. Use it to determine how solutions behave as t rightarrow infinity. All solutions eventually have positive slopes, and hence increase without bound. All solutions converge to the function y_0(t) = 0. All solutions eventually have negative slopes, and hence decrease without bound. All solutions converge to the function y_0(t) = 4. The solutions are oscillatory.

Explanation / Answer

the equation is

d((1+t^2)^2 y)/dt = (1+t^2)^-1

solution is

y = (1+t^2)^-2 (tan^-1 t +C)

all solutions converge to y_0 =0

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