2.17 Exact ode \(M\left ( x,y\right ) +N\left ( x,y\right ) y^{\prime }=0\)

ode internal name "exact"

To solve an ode of the form

\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0 \tag {1}\end{equation}

If the above ODE is exact, then there it can be written as a complete differential

\begin{align} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx} & =d\phi \left ( x,y\right ) \nonumber \\ & =\frac {\partial \phi }{\partial x}\frac {dx}{dx}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}\nonumber \\ & =\frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx} \tag {2}\end{align}

Comparing (1,2) shows that

\begin{align} \frac {\partial \phi }{\partial x} & =M\tag {3}\\ \frac {\partial \phi }{\partial y} & =N \tag {4}\end{align}

But since \(\frac {\partial ^{2}\phi }{\partial y\partial x}=\frac {\partial ^{2}\phi }{\partial x\partial y}\) then this implies

\begin{align*} \frac {\partial }{\partial y}\left ( \frac {\partial \phi }{\partial x}\right ) & =\frac {\partial }{\partial x}\left ( \frac {\partial \phi }{\partial y}\right ) \\ \frac {\partial M}{\partial y} & =\frac {\partial N}{\partial x}\end{align*}

If the above condition is satisfied, then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not satisfied then this method will not work and we have to now look for an integrating factor to force this condition, which might or might not exist. Given the ode is exact, then integrating (3) gives

\begin{equation} \phi =\int Mdx+f\left ( y\right ) \tag {5}\end{equation}

Where \(f\left ( y\right ) \) is arbitrary function to be found. Taking derivative of the above w.r.t. \(y\) gives

\[ \frac {\partial \phi }{\partial y}=\frac {d}{dy}\int Mdx+f^{\prime }\left ( y\right ) \]

Comparing the above to (4) gives an equation to solve for \(f\)

\begin{equation} \left ( \frac {d}{dy}\int Mdx\right ) +f^{\prime }\left ( y\right ) =N \tag {6}\end{equation}

Once \(f\left ( y\right ) \) is found then from (5) and since \(\phi \) is constant it becomes

\[ c=\int Mdx+f\left ( y\right ) \]

This is an implicit solution for \(y\left ( x\right ) \).