Question

The general solution of the equation $\ln \left(\frac{dy}{dx}\right)=ax+by$ is:

• A. $\frac{e^{by}}{b}=\frac{e^{ax}}{a}+C$
• B. $\frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+C$
• C. $\frac{e^{-by}}{a}=\frac{e^{ax}}{b}+C$
• D. $\frac{e^{-ay}}{a}=\frac{e^{bx}}{b}+C$

Answer 1

The correct option is B $\frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+C$

The given differetial equation can be rewritten as:
$\frac{dy}{dx}=e^{ax+by}=e^{ax}{e}^{by}$
$\Rightarrow e^{-by}dy=e^{ax}dx$
$\Rightarrow \int e^{-by}dy=\int e^{ax}dx$
$\Rightarrow \frac{e^{-by}}{-b}=\frac{e^{ax}}{a}+C$