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Solved: In each of 22 through 25, draw a direction field for the given differential
Chapter 1, Problem 23(choose chapter or problem)
In each of Problem, draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as \(t \rightarrow \infty\). If this behavior depends on the initial value of y at \(t=0\), describe this dependency. Note that the right-hand sides of these equations depend on t as well as y; therefore, their solutions can exhibit more complicated behavior than those in the text.
\(y^{\prime}=e^{-t}+y\)
Questions & Answers
QUESTION:
In each of Problem, draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as \(t \rightarrow \infty\). If this behavior depends on the initial value of y at \(t=0\), describe this dependency. Note that the right-hand sides of these equations depend on t as well as y; therefore, their solutions can exhibit more complicated behavior than those in the text.
\(y^{\prime}=e^{-t}+y\)
ANSWER:Step 1 of 3
Consider the following differential equation:
\(y' = {e^{ - t}} + y\)
The objective is to draw a direction field for the given differential equation and determine the behavior of y as \(t \to \infty\)
Rewrite the given differential equation as follows:
\(\frac{{dy}}{{dt}} = {e^{ - t}} + y\)
To sketch the direction field of the given differential equation, take some coordinate points (t,y) on the ty-plane to draw some tangent line in the direction field.