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I'm having a bit of trouble understanding the types of differential equation. I've figured out that for separable differential equations, these equations can either be linear differential equation or non- linear; however, does this apply to exact differential equation as well? (i.e Some exact differential equations are linear, whereas some are non-linear differential equation?) PS. So far I haven't seen any linear exact differential equations

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  • $\begingroup$ They can be linear (note that for instance all separable equations are exact, and homogeneous linear equations are separable) but then the usual "exact method" of computing a potential function is not necessary. $\endgroup$ – Ian Jun 21 at 1:24
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PS. So far I haven't seen any linear exact differential equations

The ode $\frac{dy}{dx}+\frac{y}{x}=0$ is linear.

Writing it as $ydx+xdy=0$. Comparing to standard form $Mdx+Ndy=0$, then $M=y,N=x$. Checking if exact

\begin{align*} \frac{dM}{dy} & =1\\ \frac{dN}{dx} & =1 \end{align*}

Since $\frac{dM}{dy}=\frac{dN}{dx}$ then the linear ODE above is exact. Note also, that if an ODE is not exact, it can still be made exact if one can find an integrating factor.

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  • $\begingroup$ That ODE is also variables-separable, so it hits the trifecta. $\endgroup$ – Gerry Myerson Jun 21 at 4:05

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