Differential equation- unfamiliar forms

**kingkaisai2** · September 1st 2006, 06:58 PM

a. 4+xdy/dx=y^2
b. e^ydy/dx-1=inx
c.ycosxdy/dx=2-ydy/dx

It would be fine if you just do one- but if anyone is feeling bored-

**Soroban** · September 1st 2006, 07:13 PM

Hello, kingkaisai2!

These are all separable . . .

$a)\;\; 4 + x\frac{dy}{dx}\:=\:y^2$

We have: . $x\frac{dy}{dx}\:=\:y^2 - 4\quad\Rightarrow\quad \frac{dy}{y^2 - 4} \:=\:\frac{dx}{x}$

$b)\;\; e^y\frac{dy}{dx} - 1 \:=\:\ln x$

We have: . $e^y\frac{dy}{dx} \:=\:1 + \ln x\quad\Rightarrow\quad e^y\,dy \:=\<img src=$ 1 + \ln x)\,dx" alt="e^y\frac{dy}{dx} \:=\:1 + \ln x\quad\Rightarrow\quad e^y\,dy \:=\

1 + \ln x)\,dx" />

$c)\;\;y\cos x\frac{dy}{dx} \:=\:2 - y\frac{dy}{dx}{$

We have: . $y\cos x\frac{dy}{dx} + y\frac{dy}{dx}\:=\:2$

Factor: . $y\frac{dy}{dx}(\cos x + 1) \:=\:2\quad\Rightarrow\quad y\,dy \:=\:\frac{2\,dx}{\cos x + 1}$

And the rest is "only Calculus" . . . LOL!

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