Question

Form a differential equation by eliminating the arbitrary constant a from the equation. y = a sin(2x + 3).??

Answer 1

`y' = 2 y cot(2x+3)`

Explanation:

`y = a sin(2x + 3) qquad bbbA`

Differentiate both sides wrt `x`:

`y' = 2a cos(2x + 3) qquad bbbB`

Use `bbbA/bbbB` to eliminate `a`

`bbbA/bbbB implies y/(y') = 1/2 tan(2x+3) qquad y' = 2 y cot(2x+3)`

That's a first-order linear ordinary differential equation, solvable by separation of variables.

Answer 2

`dy/dx=2y\cot(2x+3)`

Explanation:

Given equation:

`y=a\sin(2x+3)`

differentiating above equation w.r.t. `x` as follows

`dy/dx=\frac{d}{dx}(a\sin(2x+3))`

`dy/dx=a\frac{d}{dx}(\sin(2x+3))`

`dy/dx=a\cos(2x+3)\frac{d}{dx}(2x+3)`

`dy/dx=a\cos(2x+3)(2)`

`dy/dx=2a\cos(2x+3)`

setting `a=y/\sin(2x+3)` in above D.E. as follows

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

`dy/dx=2y\cot(2x+3)`