Pls solve this sum
​Q11. Solve the differential equation : log 5 $\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) =\mathrm{ax} +\mathrm{by}$.

 

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} {\log }_5\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)=\mathrm{ax}+\mathrm{by} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=5^{\mathrm{ax}+\mathrm{by}} \\ \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=5^{\mathrm{ax}}.5^{\mathrm{by}} \\ \Rightarrow \frac{\mathrm{dy}}{5^{\mathrm{by}}}=\frac{\mathrm{dx}}{5^{\mathrm{ax}}} \\ \Rightarrow \int \frac{\mathrm{dy}}{5^{\mathrm{by}}}=\int \frac{\mathrm{dx}}{5^{\mathrm{ax}}} \\ \Rightarrow \int 5^{-\mathrm{by}}\mathrm{dy}=\int 5^{-\mathrm{ax}}\mathrm{dx} \\ \Rightarrow \frac{5^{-\mathrm{by}}}{-\mathrm{b}}=\frac{5^{-\mathrm{ax}}}{-\mathrm{a}}+\mathrm{C} \\ \Rightarrow \frac{5^{-\mathrm{ax}}}{\mathrm{a}}-\frac{5^{-\mathrm{by}}}{\mathrm{b}}=\mathrm{C} \end{gathered}$$

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