# find in implicit form the general solution of the differential equation dy/dx = 2y^1/2(2e^2x - 5) / 3(e^2x - 5x)^2/3 (y > 0)

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### 4 Answers

You need to divide by `2y^(1/2)` both sides and to multiply by dx both sides to separate the variables x and y such that:

`(dy)/(2sqrty) = ((2e^2x - 5)dx)/(3(e^2x - 5x)^(2/3))`

You need to integrate both sides to find the general form of differential equation such that:

`int (dy)/(2sqrty) = int ((2e^2x - 5)dx)/(3(e^2x - 5x)^(2/3))`

Notice that `1/(2sqrty) = (sqrt y)'` and `(2e^2x - 5)/(3(e^2x - 5x)^(2/3)) = (root(3)(e^2x - 5x))'` Hence, you need to substitute `(sqrt y)' ` for `1/(2sqrty)` and `(root(3)(e^2x - 5x))'` for `(2e^2x - 5)/(3(e^2x - 5x)^(2/3))` such that:

`int (sqrt y)' = int (root(3)(e^2x - 5x))'`

`sqrt y = root(3)(e^2x - 5x)) + c`

You need to raise to square to remove the square root such that:

`y = root(3)((e^2x - 5x)^2) + c`

**Hence, evaluating the general solution to differential equation yields `y = root(3)((e^2x - 5x)^2)` + c.**

From here find the corresponding particular solution (in implicit form) that satisfies the initial condition y =1 when x = 0. From the original question showing full workings. I know that the value of the constant is -1 but I am not too sure

whoops that should have read the right hand side =3(e^(2x) - 5x)^(1/3). Therefore the whole answer is wrong.

The right hand side actually = 3(2^(2x) - 5x)^(1/3), which makes the whole answer wrong