Can Separation of Variables Solve My Math Problems?

[prace]

http://album6.snapandshare.com/3936/45466/844588.jpg

 

[courtrigrad]

$$\frac{dy}{dx} = e^{3x} \times e^{2y}$$ So $$\int \frac{dy}{e^{2y}} = \int e^{3x} dx$$.

 

[prace]

but the original problem had e raised to the 3x + 2y. The only way I know to get rid of that is to take the natural log of both sides right? If you do that, you are left with ln (dy/dx) on the left side. How did you get rid of the natural log there?

 

[courtrigrad]

By the properties of exponents we know that $$e^{3x+2y} = e^{3x}\times e^{2y}$$. So we can separate variables without taking the natural log of both sides. In general, $$a^{n+m} = a^{n}\times a^{m}$$

 

[prace]

oOh... I can't believe I didn't see that! sigh... it is those little things that get me all the time.

thank you so much for your help