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What is a solution to the differential equation #e^ydy/dt=3t^2+1#?

Answer 1

Elijah Abram

#y = log_e(t^3+t+C)#

#e^ydy/dt=3t^2+1# grouping variables #e^y dy=(3t^2+1)dt->e^y = t^3+t+C#

Finally

#y = log_e(t^3+t+C)#

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Answer 2

Emma Aldridge

To solve the differential equation (\frac{dy}{dt} e^y = 3t^2 + 1), separate the variables by multiplying both sides by (dt) and dividing both sides by (e^y), which gives (\frac{dy}{e^y} = (3t^2 + 1) dt). Next, integrate both sides with respect to their respective variables. The left side integrates to (-e^{-y}) and the right side integrates to (t^3 + t + C), where (C) is the constant of integration. Solving for (y), you get (y = -\ln|t^3 + t + C| + D), where (D) is another constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.