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)

Expert Answers
sciencesolve eNotes educator| Certified Educator

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.

statto45 | Student

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

statto45 | Student

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

statto45 | Student

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