How Do You Solve the Differential Equation to Find When the Population Doubles?

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The discussion focuses on solving a differential equation related to population growth, specifically \(\frac{dy}{dt}=[0.5+\sin(t)]\frac{y}{5}\) with the initial condition y(0)=1. The user is attempting to find the time \(\tau\) when the population doubles but struggles with the resulting equation \(2\cos(t)-t=10\ln(2)-2\). A key point raised is the importance of including an integration constant when solving differential equations, which the user's calculator did not account for. This oversight may be causing difficulties in determining the correct time for population doubling. Understanding the role of the integration constant is crucial for accurately solving the equation.
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Homework Statement



A certain population has a growth rate that satisfies the differential equation:

\frac{dy}{dt}=[0.5+Sin(t)]\frac{y}{5}

If y(0)=1 find the time \tau that the population has doubled.

Homework Equations


The Attempt at a Solution



This is a simple separable differential equation but when I try to solve for t when the population has doubled I get the following equation, and I can't figure out how to solve for t. I know that the equation is correct because my calculator solved it and gave the correct answer. I'm just trying to figure out where to go from here:

2Cos(t)-t=10Ln(2)-2

Thanks

Sorry about my poor use of latex
 
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Your calculator omitted an integration constant. You'll have to have talk with it. Seriously, when you integrate something it introduces an arbitrary constant, remember?
 

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