Solve the following differential equation and write the answer in the least simplified form. `y' + y = y^{5/2}`

Solve the following differential equation and write the answer in the least simplified form.

`y' + y = y^{5/2}`

Solution:

To solve the differential equation `y' + y = y^{5/2}`, we can separate the variables and then integrate them.

Separating the variables, we have:

`{y'}/{y^(5/2) }+ {1}/{y^(5/2)} = 1`

Now, integrating both sides with respect to 𝑦, we obtain:

`∫{y'}/{y^(5/2) }dy + ∫{1}/{y^(5/2)} dy = ∫1 dy`

Integrating the left-hand side, we get:

`-2/3y^(-3/2) + -2/3 = y`

Combining the terms, we have:

`-2/3y^(-3/2) - 2/3 = y` 

or 

`y = -2/3y^(-3/2) - 2/3`

This is the solution to the given differential equation in its least simplified form.