1. ## Separation of Variables

2. Like the title suggests: separate the variables.
1. $\displaystyle dr= b(cos \theta dr+ r sin \theta d\theta)= b cos \theta dr+ br sin \theta d\theta$
$\displaystyle (1- bcos \theta)dr= br sin \theta d\theta$
$\displaystyle \frac{dr}{r}= \frac{b sin \theta}{1- bcos \theta} d\theta$
Integrate.

2. $\displaystyle tan^2 y dy+ sin^3 x dx$
This isn't an equation! Is it supposed to be equal to 0? If so $\displaystyle tan^2 y dy+ sin^3 x dx= 0$ is the same as $\displaystyle tan^2 y dy= -sin^3 x dx$.
Integrate.

3. $\displaystyle dx= t(1+ t^2) sec^2 x dt$
$\displaystyle \frac{dx}{sec^2 x}= cos^2 x dx= t(1+ t^2)dt$
Integrate.

4. $\displaystyle x dx- \sqrt{a^2- x^2}dy= 0$
$\displaystyle x dx= \sqrt{a^2- x^2}dy$
$\displaystyle \frac{x}{a^2- x^2}dx= dy$
Integrate.

5. $\displaystyle x dx+ \sqrt{a^2- x^2}dy= 0$
Basically the same as 4.

6. $\displaystyle a^2 dx= x \sqrt{a^2 - x^2} dy$
$\displaystyle \frac{a^2 dx}{x\sqrt{a^2- x^2}} = dy$
Integrate.

3. ## Thanks.

many thanks. i will try to solve it.