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

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SUMMARY

The discussion focuses on solving the differential equation for population growth given by \(\frac{dy}{dt}=[0.5+\sin(t)]\frac{y}{5}\) with the initial condition \(y(0)=1\). The goal is to find the time \(\tau\) when the population doubles. The user encountered difficulty in solving the equation \(2\cos(t)-t=10\ln(2)-2\) for \(t\). A key point raised is the importance of including an integration constant when solving differential equations, which the user's calculator omitted.

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Homework Statement



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

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

If y(0)=1 find the time [tex]\tau[/tex] 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
 
Physics news on Phys.org
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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