MHB Indie's question at Yahoo Answers regarding a separable ODE/partial fractions

AI Thread Summary
The discussion focuses on solving a first-order separable ordinary differential equation (ODE) given by dy/dx = 0.3y - 0.0001y². The solution process involves eliminating decimals by multiplying through by 10,000, followed by separating variables and applying partial fraction decomposition. The decomposition leads to the integral of the left side resulting in a logarithmic equation. Finally, the solution is expressed in terms of y(x) = 3000 / (1 + Ce^(-3/10 x)), providing a clear method for solving similar ODEs. This approach is particularly useful for students in BC Calculus seeking to understand separable equations and partial fractions.
MarkFL
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Here is the question:

Partial Decomposition Problem (Calculus)?

I am doing some review problems but I don't understand how to do this problem... any suggestions would be helpful. Thank you in advance. This is for my BC Calculus Class in Nebraska so not too many people are familiar with this math here that I know.

(dy/dx) = .3y - .0001(y^2)

I have posted a link there to this thread so the OP can view my work.
 
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Hello Indie,

We are given to solve the first order ODE:

$$\frac{dy}{dx}=0.3y-0.0001y^2$$

First, let's multiply through by $10000$ to get rid of the decimals:

$$10000\frac{dy}{dx}=3000y-y^2$$

Next, let's separate variables:

$$\frac{1}{y(3000-y)}\,dy=\frac{1}{10000}\,dx$$

Now, the expression on the left may be rewritten using partial fraction decomposition. We may assume the decomposition will take the form:

$$\frac{1}{y(3000-y)}=\frac{A}{y}+\frac{B}{3000-y}$$

Multiplying through by $$y(3000-y)$$ we obtain:

$$1=A(3000-y)+By$$

Since this is true for all $y$, letting $y=0$ we get:

$$1=3000A\implies A=\frac{1}{3000}$$

And letting $y=3000$, we obtain:

$$1=3000B\implies B=\frac{1}{3000}$$

Hence:

$$\frac{1}{y(3000-y)}=\frac{1}{3000}\left(\frac{1}{y}+\frac{1}{3000-y} \right)$$

And so the ODE may be written:

$$\left(\frac{1}{y}-\frac{1}{y-3000} \right)\,dy=\frac{3}{10}\,dx$$

Integrating, we find:

$$\int \frac{1}{y}-\frac{1}{y-3000}\,dy=\frac{3}{10}\int\,dx$$

$$\ln\left|\frac{y}{y-3000} \right|=\frac{3}{10}x+C$$

Converting from logarithmic to exponential form (and rewriting the parameter $C$), we have:

$$\frac{y}{y-3000}=Ce^{\frac{3}{10}x}$$

Now we want to solve for $y$. Multiply through by $y-3000$:

$$y=(y-3000)Ce^{\frac{3}{10}x}$$

$$y\left(Ce^{\frac{3}{10}x}-1 \right)=3000Ce^{\frac{3}{10}x}$$

$$y=\frac{3000Ce^{\frac{3}{10}x}}{Ce^{\frac{3}{10}x}-1}$$

Now, dividing each term in the numerator and denominator by $$Ce^{\frac{3}{10}x}$$ and rewriting the parameter again, we obtain:

$$y(x)=\frac{3000}{1+Ce^{-\frac{3}{10}x}}$$
 
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