Solve the given differential equation by separation of variables. `dy/dx = ((2y+3)/(4x+5))^2`

Solve the given differential equation by separation of variables.

`dy/dx = ((2y+3)/(4x+5))^2`

Solution:

Start by isolating `dy` on one side of the equation:

`dy = ((2y+3)/(4x+5))^2 dx`

Next, divide both sides of the equation by `((2y+3)/(4x+5))^2` to get:

`(dy/((2y+3)/(4x+5))^2) = dx`

Now, we can separate the variables by moving `dx` to one side and `dy` to the other:

`(1/((2y+3)/(4x+5))^2) dy = dx`

Integrate both sides with respect to the isolated variable:

`∫(1/((2y+3)/(4x+5))^2) dy = ∫dx`

Solving the integral on the left side:

`-1/(2y+3) = x + C`

Solving the integral on the right side:

`x = x + C`

Now we can solve for y by isolating y on one side:

`-1/(2y+3) = x + C`

`y = (-1/(2x+2C+3))`