Question

Find the general solution of the given differential equation. cos^2(x)sin(x)dy/dx+(cos^3(x))y=1 g

Answer 1

If the given differential equation is

`\cos^2(x) \sin(x) (dy)/(dx) + \cos^3(x) y = 1`

then multiply both sides by
`\frac1{\cos^2(x)}` :

`\sin(x) (dy)/(dx) + \cos(x) y = \sec^2(x)`

The left side is the derivative of a product,

`(d)/(dx)\left[\sin(x)y\right] = \sec^2(x)`

Integrate both sides with respect to
`x`, recalling that
`(d)/(dx)\tan(x) = \sec^2(x)` :

`\displaystyle \int (d)/(dx)\left[\sin(x)y\right] \, dx = \int \sec^2(x) \, dx`

`\sin(x) y = \tan(x) + C`

Solve for
`y` :

`\boxed{y = \sec(x) + C \csc(x)}</p><p>which follows from [tex]\tan(x)=(\sin(x))/(\cos(x))`.