Separation of Variables Help

prace

courtrigrad

[tex] \frac{dy}{dx} = e^{3x} \times e^{2y} [/tex] So [tex] \int \frac{dy}{e^{2y}} = \int e^{3x} dx [/tex].

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 [tex] e^{3x+2y} = e^{3x}\times e^{2y} [/tex]. So we can separate variables without taking the natural log of both sides. In general, [tex] a^{n+m} = a^{n}\times a^{m} [/tex]

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