The discussion focuses on the conditions under which an ordinary differential equation (ODE) can be solved as an exact equation. It establishes that for an ODE in the form M(x,y)dx + N(x,y)dy = 0, where M = ∂F/∂y and N = ∂F/∂x, the equation is only exact if the condition ∂N/∂x = ∂M/∂y holds true. This condition is critical for determining the solvability of the ODE using methods applicable to exact differential equations.
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AlonsoMcLaren said:If an ODE can be written in the form M(x,y)dx+N(x,y)dy=0, where M=δf/δy and N=δf/δx. Why isn't it correct to solve the ODE using the way analogous to the way solving Exact differential equations?