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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.

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

Differential Equations

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Missouri State University

Harvey Mudd College

Idaho State University

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