solve the differential equation y = 3^x. logarithmic functions

Explanation:

Step1: Recall antiderivative rule

The antiderivative of (a^{x}) is (\frac{a^{x}}{\ln a}+C) ((a>0,a\neq1)). Here (a = 3).

Step2: Integrate both sides

Since (y'=\frac{dy}{dx}=3^{x}), then (y=\int 3^{x}dx). Using the formula (\int a^{x}dx=\frac{a^{x}}{\ln a}+C), we have (y=\frac{3^{x}}{\ln 3}+C).

Answer:

(y=\frac{3^{x}}{\ln 3}+C)