# Find: Use integration to find general solution of the differential equation \frac{dy}{dx}= \sec...

## Question:

Use integration to find general solution of the differential equation

{eq}\frac{dy}{dx}= \sec 3x \tan 3x {/eq}

## Solving Differential Equation with Separable Variables:

It is said that the differential equation of form {eq}\frac{dy}{dx}=\frac{g(x)}{h(y)} {/eq} is separable, or has separable variables.

The above equation can be written in the form {eq}h(y)dy=g(x)dx {/eq}, which indicates the procedure to solve separable differential equations. By integrating both members of {eq}h(y)dy=g(x)dx {/eq} the general solution is obtained, which is generally expressed implicitly.

## Answer and Explanation:

We need to solve {eq}\frac{dy}{dx}= \sec 3x \tan 3x {/eq}.

If we separate the variables:

{eq}dy= \sec 3x \tan 3x dx {/eq}

Integrating:

{eq}\int dy= \int \sec 3x \tan 3x dx\\ y=\frac{1}{3}\sec 3x+C\space{}\text{[General solution]} {/eq}