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Alex17521 [72]
2 years ago
6

Does anyone know how to do this

Mathematics
1 answer:
dusya [7]2 years ago
5 0

\dfrac{\mathrm dy}{\mathrm dx}=(y+2)^2\sin\left(\dfrac xe\right)

a. If y=k, then \frac{\mathrm dy}{\mathrm dx}=0, so

0=(k+2)^2\sin\left(\dfrac xe\right)\implies (k+2)^2=0\implies k=-2

b. When x=w, we're told y has a horizontal tangent, which has slope \frac{\mathrm dy}{\mathrm dx}=0. So we have

0=(y+2)^2\sin\left(\dfrac we\right)\implies\sin\left(\dfrac we\right)=0\implies\dfrac we=2n\pi\implies w=2ne\pi

where n is any integer, whose smallest positive value occurs for n=1, giving w=2e\pi.

c. The equation is separable:

\dfrac{\mathrm dy}{(y+2)^2}=\sin\left(\dfrac xe\right)\,\mathrm dx

Integrate both sides to get

-\dfrac1{y+2}=-e\cos\left(\dfrac xe\right)+C

y=-1 when x=0, so we find

-1=-e+C\implies C=e-1

Then the particular solution to the DE is

-\dfrac1{y+2}=-e\cos\left(\dfrac xe\right)+e-1\implies y=\dfrac1{e\left(\cos\left(\frac xe\right)-1\right)+1}-2

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