Verify that the family of functions `y(x)=-5/3x^3+c, c in RR` is the general solution of the equation `y'=-5x^2` . Graph the general solution for each value of `c` and the particular solution `y(x)=-5/3x^3+1` .

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To verify that every function of the form `y(x)=-5/3x^3+c` is a solution, we need to differentiate. Indeed, it follows immediately by the Power Rule and the Constant Multiple Rule that

`y'(x)=-5x^2` for any value of `c.`

Furthermore, these are the only solutions because if two functions have the same derivative, they can differ only by a constant. This proves that the solution set of `y'(x)=-5x^2` is precisely `{y(x)=-5/3x^3+c : cinR}.`

It's not a good idea to graph the solution for *each *value of `c` (you could but they would fill the whole plane), but here are several. The black graph is the particular solution `y(x)=-5/3x^3+1.`

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