What is the Unique Solution Region for (1+y^3)y' = x^2 in the xy-plane?

[robierob12]

Determine the region in the xy-plane for which

(1+y^3)y' = x^2

This has a unique solution.

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Not really understanding what this is asking of me?

I solved this as a seperable diff. Equation.

(1+ y^3)dy = (x^2) dx

y + (1/4)y^4 = (1/3) x^3 + c

3y + (3/4)y^4 = x^3


not sure what this one wants from me...

the general solution to which all the solution curves can be reached from?


any help is appreciated.

Rob

 

[EnumaElish]

The solution you obtained is a polynomial in y. Which typically means that there are multiple y functions (roots) that solve the polynomial. The question is, is there a subset of the xy plane over which all roots are identical.

 

[HallsofIvy]

Determine the region in the xy-plane for which

(1+y^3)y' = x^2

This has a unique solution.

-----------------------------------------

Not really understanding what this is asking of me?

I solved this as a seperable diff. Equation.

(1+ y^3)dy = (x^2) dx

y + (1/4)y^4 = (1/3) x^3 + c

3y + (3/4)y^4 = x^3

What happened to the "c"?? If you were to solve for y, since this is a fourth degree equation, you might get 4 separate solutions. For what values of c does this equation have a unique solution? What initial (x,y) values give those values of c?

not sure what this one wants from me...

the general solution to which all the solution curves can be reached from?


any help is appreciated.

Rob