Question

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

Explanation:

Step1: Recall antiderivative rule

The antiderivative of \(a^{x}\) is \(\frac{a^{x}}{\ln a}+C\) (\(a>0,a
eq1\)). 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:

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