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

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In summary, the conversation discusses a population growth rate differential equation and finding the time when the population doubles. The attempt at a solution involves solving for t but the calculator's solution is missing an integration constant.
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sammyb787
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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
 
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  • #2
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?
 

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

How do I solve a simple algebraic equation?

To solve a simple algebraic equation, you need to isolate the variable on one side of the equal sign by using inverse operations. This means performing the opposite operation to both sides of the equation until the variable is left alone on one side.

What is the order of operations when solving an equation?

The order of operations is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This means that you should always solve operations within parentheses first, then exponents, followed by multiplication and division, and finally addition and subtraction.

How do I know if my solution is correct?

You can check your solution by substituting the value you found for the variable back into the original equation and seeing if both sides are equal. If they are, then your solution is correct.

What do I do if there are variables on both sides of the equation?

If there are variables on both sides of the equation, you need to first simplify each side by combining like terms. Then, you can use inverse operations to isolate the variable on one side of the equation.

Can I solve an equation without using inverse operations?

No, inverse operations are necessary to solve an equation. This is because equations are based on the principle of balance, and using inverse operations maintains this balance and allows you to solve for the variable.

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