22-06-2024 18:46

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Tags : mathematics Differential Equations

Exact Linear Differential Equations

Definition :

Differential equations are often in the form of

$$M(x,y)dx+N(x,y)dy=0$$

Two Variables For one variable refer Basic Undergraduate Differential Equations

If this Differential Equation is Exact only if there is a function of two variables $u(x,y)suchthatd(u)=Mdx+Ndy$

Where $u(x,y)=C$ where C is the arbitrary constant

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Condition for differential equation to be exact :

The differential equation $M(x,y)dx+N(x,y)dy=0$ Where $M$ And $N$ are functions of $x$ and $y$ Will be [[Exact Differential Equation|Exact differential equation]] When $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$

And the solution will be given by ${\int }_{\mathrm{y}\mathrm{constant}}Mdx+\int \left(\mathrm{terms}\mathrm{of}\mathrm{N}\mathrm{free}\mathrm{from}x\right)dy=c$

If your equation is not reduceable then check Equations Reducible to exact equations

References