How to Solve Basic Algebra Questions: Tips and Tricks

[kape]

I have a few nagging questions that are preventing me from solving calculus problems.. Can someone give me a hand?


----------
Question 1

$$e^xy' = (4x+1)y^2$$

From the equation above, is it possible to do this:

$$\frac{y'}{y^2} = \frac{4x+1}{e^x}$$

Aren't you supposed to divide one at a time, like this?

$$\frac{e^xy'}{y^2} = (4x+1)$$

How would I get the $$\inline e^x$$ to the other side?


----------
Question 2

How do you tell the difference between these two?

$$2 - 3 = -1$$ and $$2-3 = -6$$ ?

(One latter being (2)(-3) = - 6?)

 

[assyrian_77]

Q1. Yes, it is possible. How to get the $e^x$ to the other side? Just divide both sides by $e^x$. Which is the same as your first statement.

Q2. What do you mean? One DOES NOT write $2\cdot(-3)$ as $2-3$!

 

[HallsofIvy]

There is, in fact, no "law" against dividing by several things at the same time: you could go from e^xy'= (4x+ 1)y² to
$$\frac{y'}{y^2} = \frac{4x+1}{e^x}$$
by dividing both sides by e^xy²
OR by first dividing by y² and THEN dividing by e^x. The math doesn't care.

 

[J77]

I guess the dash represents differentiating w.r.t x...

$$e^x \frac{dy}{dx}=(4x+1)y^2$$

$$\frac{1}{y^2}\frac{dy}{dx}=\frac{(4x+1)}{e^x}$$

$$\int_y{y^{-2}}dy=\int_x{(4x+1)e^{-x}dx$$