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Numerade Educator

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Problem 2 Medium Difficulty

A direction field for the differential equation $ y^1 = x \sin y $ is shown.
(a) Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) $y(0)=1$
(ii) $y(0)=0.2$
(iii) $y(0)=2$
(iv) $y(1)=3$
(b) Find all the equilibrium solutions.

Answer

a) (iii) $y(0)=2$ The curve must pass through point $(0,2) .$ The slope line segments are all horizontal at $y=2,$ and $y^{\prime}=0$ at $y=2$ so this is a horizontal line.
$\quad(\mathrm{iv}) y(1)=3$
The curve must pass through point $(1,3) .$
(iv) $y(1)=3$
The curve must pass through point $(1,3)$
b)
Equilibrium solutions are $y=$ Any even number

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Video Transcript

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