Integrating differential equations

Question

This is the question:

$$dB/(x-yB)=dt$$

After trying several times to solve this, this is what I've come up with:

$$B(t)=xB/y + C(\exp[-ty]), where C=\exp(c)$$

I have a feeling I've made a mistake along the way, and would appreciate it if someone could give me a few pointers!

Thank you :)

Answer 1

$$\frac{1}{ax-yb}\space\text{d}b=\text{d}t\Longleftrightarrow\int t'(b)\space\text{d}b=\int\frac{1}{ax-yb}\space\text{d}b$$

Use:

1. $$\int t'(b)\space\text{d}b=t(b)+\text{C}$$
2. Substitute $u=ax-by$ and $\text{d}y=-y\space\text{d}b$ $$\int\frac{1}{ax-yb}\space\text{d}b=-\frac{1}{y}\int\frac{1}{u}\space\text{d}u=-\frac{\ln\left|u\right|}{y}+\text{C}=\text{C}-\frac{\ln\left|ax-by\right|}{y}$$