Given a first-order differential equation
of the form \(\dot{y} = f(t,y),\)
sketch the direction field for that differential equation,
and use the direction field to sketch the curve
of an approximate particular solution given an initial condition.
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Consider the differential equation \(\dot y = t-\frac{1}{2}y\).
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Sketch a direction field
for this differential equation
for \({-6 \lt y \lt 6}\) and \({-6 \lt t \lt 6\,.}\)
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Using your direction field,
sketch the curves of the particular solutions
that correspond to each of these initial conditions:
\[ y(0) = 1 \qquad\qquad y(-3) = 0 \qquad\qquad y(2) = -1\,.\]
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Consider the differential equation \(\dot y = \frac{1}{4}t^2+\frac{1}{4}y^2-3\,.\)
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Sketch a direction field
for this differential equation
for \({-6 \lt y \lt 6}\) and \({-6 \lt t \lt 6\,.}\)
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Using your direction field,
sketch the curves of the particular solutions
that correspond to each of these initial conditions:
\[ y(0) = 0 \qquad\qquad y(0) = 3 \qquad\qquad y(3) = 3\,.\]