Question

Y' = t⁻¹ - y². Describe how the solutions appear to behave as t increases and how their behavior depends on the initial value y when t = 0.

a) Solutions approach infinity as t increases; behavior depends on the sign of y at t = 0.
b) Solutions approach zero as t increases; behavior depends on the magnitude of y at t = 0.
c) Solutions stabilize around a constant value as t increases; behavior is independent of the initial value y.
d) Solutions oscillate periodically as t increases; behavior depends on the parity of y at t = 0.

Answer 1

The solutions to the given equation y' = t⁻¹ - y² approach zero as t increases, and their behavior depends on the magnitude of y at t = 0.

Step-by-step explanation:

The given equation is y' = t⁻¹ - y². To understand how the solutions behave as t increases and how their behavior depends on the initial value y when t = 0, we can analyze the equation. In this case, the equation represents a first-order ordinary differential equation.

By examining the equation, we can see that as t approaches infinity, the term t⁻¹ becomes negligible compared to y². Therefore, the equation can be simplified to y' = -y².

This differential equation represents exponential decay. The solutions will approach zero as t increases, regardless of the sign of y at t = 0. Therefore, the correct answer is b) Solutions approach zero as t increases; behavior depends on the magnitude of y at t = 0.