Question

Solve the differential equation log(dy/dx)=ax+by​

Answer 1

Step-by-step explanation:

plaease check the image

then solve the integration

Answer 2

Therefore the required solution is

$\frac{e^{ax}}{a}+ \frac{e^{-by}}{b}+C=0$

Step-by-step explanation:

Formula:

$\int e^{mx}dx=\frac{e^{mx}}{m} +C$

Given that,

$log(\frac{dy}{dx} )=ax+by$

$\Rightarrow \frac{dy}{dx} =e^{ax+by}$

$\Rightarrow \frac{dy}{dx} = e^{ax}.e^{by}$

$\Rightarrow \frac{dy}{e^{by}} =e^{ax}{dx}$

Integrating both sides,

$\Rightarrow \int\frac{dy}{e^{by}} =\int e^{ax}{dx}$

$\Rightarrow \int e^{-by}dy =\int e^{ax}{dx}$

$\Rightarrow \frac{e^{-by}}{-b} =\frac{e^{ax}}{a} +C$      [ where c is arbitrary constant]

$\Rightarrow \frac{e^{ax}}{a}+ \frac{e^{-by}}{b}+C=0$

Therefore the required solution is

$\frac{e^{ax}}{a}+ \frac{e^{-by}}{b}+C=0$