Exact differential equations · Differential Equations

Intuition

Some first-order equations are what you get by differentiating a relation. If F(x, y) = C describes a curve, differentiating along it gives an equation whose solutions are exactly the level curves of F: the rate of change of F in x, plus its rate of change in y times the slope, is zero. An exact equation is one that can be recognised as coming from such an F, and solving it means recovering F.

A contour map. Walking along a contour line your height never changes. An exact equation says how height changes in each direction and asks for the contours; its solutions are the contours.

plane diagram: the-solutions-are-level-curves

The equation $y\,dx+x\,dy=0$ is exact: it is what differentiating $F(x,y)=xy$ along a curve gives. Its solutions are the level curves $xy=C$ , drawn for $C=\pm1$ and $C=2.5$ .

Equations that are a derivative along a curve

Write a first-order equation as $M(x,y)\,dx+N(x,y)\,dy=0$ , which means $M+Ny'=0$ . It is exact if some function , called a potential, has and . Then along any solution , so is constant on it: the solutions are the level curves .

F_{x}=M,\quad F_{y}=N\ \Longrightarrow\ F(x,y)=C

What to know

A partial derivative differentiates in one variable holding the other fixed: for $F=x^{2}y$ , $F_{x}=2xy$ and . This, with the chain rule along a curve, is all the multivariable calculus the chapter uses, and it is taken on trust.

Level curves of a potential are solutions

\begin{aligned}&F_{x}=M,\ F_{y}=N,\ M+Ny'=0\\&\Longrightarrow\ F\bigl(x,y(x)\bigr)=C\end{aligned}

Take a solution and follow the potential along its graph. The chain rule for a function of two variables, each depending on x, says the rate of change of F along the curve is its partial in x plus its partial in y times y prime. By exactness those partials are M and N, and M plus N times y prime is zero because y solves the equation. A function with zero derivative on an interval is constant, so the graph lies on one level curve.

Proof steps

\frac{d}{dx}F\bigl(x,y(x)\bigr)=F_{x}+F_{y}\,y'

The chain rule for a function of two variables along a curve.

F_{x}+F_{y}\,y'=M+N\,y'

Exactness replaces the partial derivatives by M and N.

M+N\,y'=0

That is zero because y solves the equation.

F\bigl(x,y(x)\bigr)=C

A function with zero derivative on an interval is constant, so the solution lies on a level curve of F.

Applications

Practice

Partial Derivatives, Briefly

For a function of two variables, $F_{x}$ is its derivative in $x$ with $y$ held fixed, and $F_{y}$ its derivative in with held fixed.

F=x^{2}y+y^{3}:\quad F_{x}=2xy,\quad F_{y}=x^{2}+3y^{2}

Try it

For $F(x,y)=x^{3}y^{2}$ , what is $F_{y}$ ?