Question

How do you find all solutions to the equation `y(2x)=dy/dx` where `y` is a function of `x`?

Answer 1

Use the separation of variables method.

Explanation:

The separation of variables method is:

Perform algebraic operations so the all of the factors containing y are on the same side of the equation with `dy` and all of the factors containing x and `dx` are on the opposite side of the equation.

Given: `y(2x)=dy/dx`

Multiply both sides of the equation by `dx/y`:

`(2x)dx=dy/y`

Integrate both sides:

`int(1/y)dy = int (2x)dx`

`ln|y| = x^2+C`

Use the exponential function on both sides:

`e^(ln|y|) = e^(x^2+C)`

The left side becomes y, because the exponential and the natural logarithm are inverses:

`y = e^(x^2+C)`

Adding an arbitrary constant, C, in the exponent, is the same as multiplying by some arbitrary constant, C:

`y = Ce^(x^2)`