In each of the following problems, draw a direction field. Notice that this problem is motivated by the fact that our analytical methods for the solution of first-order differential equations do not work for these equations. Using the direction field, sketch the solutions for several possible initial values y0. Describe how solutions appear to behave as t grows, and how their behavior depends on the initial value y0. (a) y′ =ty(3−y), y(0)=y0 (b) y′ =y(3−ty), y(0)=y0 (c) y′ =−y(3−ty), y(0)=y0 (d) y′ =t−1−y2, y(0)=y0
In each of the following problems, draw a direction field. Notice that this problem is motivated by the fact that our analytical methods for the solution of first-order differential equations do not work for these equations. Using the direction field, sketch the solutions for several possible initial values y0. Describe how solutions appear to behave as t grows, and how their behavior depends on the initial value y0.
(a) y′ =ty(3−y), y(0)=y0
(b) y′ =y(3−ty), y(0)=y0
(c) y′ =−y(3−ty), y(0)=y0
(d) y′ =t−1−y2, y(0)=y0
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According to the given information, it is required to draw the direction field.
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