Differential Equations Help

1. Sep 28, 2005

amcavoy

I have the following to solve:

$$\frac{dx}{dt}=-\alpha xy;\quad y=y_0e^{-\beta t};\quad x(0)=x_0$$

I separate variables and come up with:

$$\frac{dx}{x}=-\alpha y_0e^{-\beta t}dt$$

$$\ln{x}=-\alpha y_0\int e^{-\beta t}dt=\frac{\alpha y_0}{\beta}e^{-\beta t}+C$$

...so for a final answer I come up with:

$$x=x_0\exp{\left(\frac{\alpha y_0}{\beta}e^{-\beta t}\right)}$$

..however the book says that the answer is:

$$x=x_0\exp{\left(\frac{-\alpha y_0\left(1-e^{-\beta t}\right)}{\beta}\right)}$$

I cannot find where I went wrong, any ideas?

Thanks a lot.

2. Sep 28, 2005

Tide

You didn't evaluate the time integral at both limits.

3. Sep 28, 2005

hotvette

Good catch. I stared at it for a few minutes and couldn't figure it out.

4. Sep 28, 2005

amcavoy

...both limits?

5. Sep 28, 2005

saltydog

You have:

$$x(t)=K\text{Exp}\left[\frac{\alpha y_0}{\beta}e^{-\beta t}\right]$$

with:

$$x(0)=x_0$$

Now, carefully substitute that initial value into the equation to solve for K.

6. Sep 28, 2005

amcavoy

Ahh, I must have put the x0 term there prematurely. Thanks for the help everyone, I have it now.