Solve the implicit ode by finding an integrating factor m that is a either a function m(x) of x or a function m(y) of y only. now multiply the equation by the integrating factor that you have found and then integrate the resulting equation to get a solution in implicit form.

This equation is not exact, since




So we look for an integrating factor such that



For this to be exact, we require



Differentiating both sides gives




When we take to be a function of either or , but not both, this partial differential equation reduces to one of the separable ordinary differential equations




both of which can be solved directly for provided that the result on the right hand side of either ODE is a function of either only or , respectively.

The choice of which integrating factor to look for is then decided by how easily the right hand side can be taken care of. We have



On the other hand, the integral resulting from an integrating factor is more complicated/impossible to deal with. So, the integrating factor must be a function of , which means , and it satisfies




Distributing the integrating factor across the original ODE, we have



with partial derivatives




Thus the modified ODE is exact, as required. Now we try to find a solution of the form .










Therefore the general solution is

D.

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