`dy/dx = (6-x^2)/ (2y^3)` Find the general solution of the differential equation

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`(dy)/dx = (6-x^2)/(2y^3)`

This differential equation is separable since it can be rewritten in the form

`N(y) dy=M(x) dx`

So separating the variables, the equation becomes

`2y^3dy = (6-x^2)dx`

Integrating both sides, it result to

`int 2y^3dy = int (6-x^2)dx`

`2*y^4/4 + C_1 = 6x-x^3/3 + C_2`

`y^4/2 + C_1 = -x^3/3 + 6x + C_2`

Since C1 and C2 represent any number, it can be expressed as a single constant C.

`y^4/2 = -x^3/3 + 6x + C`

 

Therefore, the general solution of the given differential equation is `y^4/2 = -x^3/3 + 6x+ C` .

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