The solution of the differential equation $log \left(\frac{dy}{dx}\right)=ax+by $ is :
Answer & explanation
Correct answer: option 3
Given: $\ln\!\left(\frac{dy}{dx}\right)=ax+by$
Hence $\frac{dy}{dx}=e^{ax+by}=e^{ax}e^{by}$
Separate variables: $e^{-by}\,dy=e^{ax}\,dx$
Integrate both sides:
$\displaystyle \int e^{-by}\,dy=\int e^{ax}\,dx$
$-\frac{1}{b}e^{-by}=\frac{1}{a}e^{ax}+C$