# Find the general solution. cos^2(x) \sin(xy') + cos^3(xy) = 1

## Question:

Find the general solution.

{eq}cos^2(x) \sin(xy') + cos^3(xy) = 1 {/eq}

## The general solution of a differential equation

A general solution is a solution of an ordinary differential equation of order n that involves exactly n essential arbitrary constants, thus also calling it a complete solution.

{eq}\cos ^{2}x\sin xy' + \cos ^{3}xy=1 {/eq}

y' + cotxy = {eq}\frac{1}{\cos ^{2}x\sin x} {/eq}

If {eq}e^{\int \cot xdx} = e^{lnsinx} = sinx {/eq}

The general solution is

{eq}y[sinx] = \int \frac{1}{\cos ^{2}x\sin x}\sin xdx {/eq}

{eq}y[sinx] = \int \sec ^{2}xdx {/eq}

{eq}y[sinx] = \tan x + C {/eq}

We have,

{eq}y = \frac{\tan x + C}{\sin x} {/eq}

First-Order Linear Differential Equations

from

Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.