How to Solve a Non-Exact Differential Equation with an Integrating Factor?

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SUMMARY

The discussion focuses on solving the non-exact differential equation (x+x^4)dy+y(y^3-x^3)dx=0. The original equation is identified as neither separable nor homogeneous, prompting the need for an integrating factor (IF). However, the integrating factors derived involve both x and y, which is not suitable. A successful approach involves a change of variables to X=x^3 and Y=y^3, transforming the problem into a Riccati ODE, facilitating a solution.

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Ptopenny
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the problem is (x+x^4)dy+y(y^3-x^3)dx=0
well I know that this is not a separable equation, homogenous equation or an exact equation...so i try to solve it by treating it as a non exact DE by finding out the integrating factor...but the both IF come out in term of x and y which involve 2 variables where by IF must only has one variable...
 
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Hi !
The pattern of the ODE makes think to a change of variables : X=x^3 and Y=y^3, which leads to a Riccati ODE.
 

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thank god...u really help me very much :D :D
 

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