Math 2.0

Ordinary differential equations Definition

Exact equation

Builds on Ordinary differential equationSolution of an ordinary differential equation

Definition

Let DD be an open subset of ℝ2\mathbb{R}^2 and let M,N:D→ℝM, N : D \to \mathbb{R} be continuous functions. The equation M(x,y)+N(x,y) y′=0,M(x, y) + N(x, y)\, y' = 0, also written M dx+N dy=0M\, dx + N\, dy = 0, is exact on DD if there is a differentiable function ψ:D→ℝ\psi : D \to \mathbb{R} with ∂ψ∂x=Mand∂ψ∂y=Non D.\frac{\partial \psi}{\partial x} = M \quad \text{and} \quad \frac{\partial \psi}{\partial y} = N \quad \text{on } D. Such a function ψ\psi is a potential function of the equation.

Remarks

Let the equation be exact. Then a differentiable function yy on an interval, with graph in DD, is a solution if and only if ψ(x,y(x))\psi(x, y(x)) is constant. This follows from the chain rule: ddxψ(x,y(x))=M+Ny′\frac{d}{dx} \psi(x, y(x)) = M + N y'. So the solutions are given implicitly by ψ(x,y)=C\psi(x, y) = C. Example: 2xy+x2y′=02 x y + x^2 y' = 0 is exact with ψ=x2y\psi = x^2 y, and its solutions satisfy x2y=Cx^2 y = C. The test for exactness decides whether an equation is exact.

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  1. Differential equation concept
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  5. Solution of an ordinary differential equation definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Exact_differential_equation, First-order exact differential equations
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DefinesExact equation
UsesOrdinary differential equation, Solution of an ordinary differential equation

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