How to find the implicit function of $\,\dfrac{\mathrm dx}{\mathrm dt}=y+y^2-x^3=y+P_2(x,y)\;\;?$
I am facing difficulty to find the implicit function in the given example (the picture is attached).
How can we find the implicit function ?
2026-03-30 11:17:05.1774869425
How to find the implicit function of $\,\mathrm dx/\mathrm dt=y+y^2-x^3=y+P_2(x,y)\;?$
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Your task is not to find some ODE solution, but the curve where the right side is zero for $x,y\approx 0$. $$ 0=y+y^2-x^3 $$ As $y^2\ll |y|$, the only balance of terms is for $0\approx y-x^3$. Now refine this first approximation by treating $$ y_{new}=x^3-y_{old}^2 $$ as fixed-point iteration $$ y=x^3-x^6\\ y=x^3-x^6+2x^9+O(x^{12})\\ y=x^3-x^6(1-x^3+2x^6)^2+O(x^{15}) $$ etc.