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garri49 [273]
3 years ago
12

What do I do with e^y in the differential equation dy/dt =(2t)/(e^y)?

Mathematics
1 answer:
Lelechka [254]3 years ago
7 0

As you said, the equation is separable:

d<em>y</em>/d<em>t</em> = 2<em>t</em> / exp(<em>y</em>)

exp(<em>y</em>) d<em>y</em> = 2<em>t</em> d<em>t</em>

Integrate both sides:

∫ exp(<em>y</em>) d<em>y</em> = ∫ 2<em>t</em> d<em>t</em>

exp(<em>y</em>) + <em>C</em>₁ = <em>t</em> ² + <em>C</em>₂

Move the constant terms to one side. When you add them together, you get another constant, so you can ignore the subscript altogether:

exp(<em>y</em>) = <em>t</em> ² + <em>C</em>

Solve for <em>y</em> explicitly by taking the logarithm of both sides:

ln(exp(<em>y</em>)) = ln(<em>t</em> ² + <em>C </em>)

<em>y</em> = ln(<em>t</em> ² + <em>C </em>)

<em>C</em> can be any number; if it happens to be 0, then you have

<em>y</em> = ln(<em>t</em> ²) = 2 ln(<em>t</em> )

so B is the correct choice.

You can also approach this from the opposite angle: Assume <em>y</em> is one of the given solutions, then substitute it into the ODE. (Bit more trial-and-error involved, so not a good idea if you're short on time.)

For example, if <em>y</em> = 2 exp(<em>t</em> ) as in choice A, you have d<em>y</em>/d<em>t</em> = 2 exp(<em>t</em> ), so the ODE would become

2 exp(<em>t</em> ) = 2<em>t</em> / exp(2 exp(<em>t</em> ))

which is clearly (I hope) not true.

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