** Problem**: Solving differential equation

** Solution**:

Let

Then by chain rule,

Rewrite (1) as

Equivalently,

Integrate (3) with respect to gives

Hence

Or,

where since .

Square (4) gives

Solving for from (5), we obtain

And so,

Or,

Integrate (6) with respect to yields

i.e.,

Therefore,

since is an even function.

Let

i.e.,

Let

By definition (see “Deriving Two Inverse Functions“),

i.e.,

From “Integration of Trigonometric Expressions ” :

Without using a CAS,

Let ,

i.e.,

*Exercise-1* What is the derivative of hint: