# Exact Form

A differential equation written in the form $M(x,y)dx+N(x,y)dy=0$ where $M$ and $N$ are functions of $x$ and $y$ or both, is said to be in an exact form if there exists a function $ƒ(x,y)$ of $x$ and $y$ such that $Mdx+Ndy=dƒ(x,y)$ i.e. when $Mdx+Ndy$ is an exact or a perfect differential.

The differential equation $ydx+xdy=0$ is exact, since $ydx+xdy=d(xy)=0$ On integration, we get $xy=C$ where $C$ is an arbitrary constant. But, the differential equation $xdy-ydx=0$ is not exact as it stands. However, it becomes exact if we multiply its both sides by $\frac{1}{x^2}$, since $\frac{xdy-ydx}{x^2}=0$ $d\left(\frac{y}{x}\right)=0$ On integration, we get $\frac{y}{x}=C$ $\text{or, }y=Cx$ An expression or factor such as $\frac{1}{x^2}$ is called an Integration Factor (I.F.).

## Solve, by reducing to exact form, the following equations:

### $ydx-xdy=xydy$

$ydx-xdy=xydy$ $\frac{dx}{x}-\frac{dy}{y}=dy$ $d(\log x)-d(\log y)=dy$ $d(\log x-\log y)=dy$ Integrating both sides, we get $\log x-\log y=y+C$ $\therefore\log\frac{x}{y}=y+C$

### $2xydx+x^2dy=0$

$2xydx+x^2dy=0$ $d(x^2y)=0$ Integrating both sides, we get, $x^2y=C$

### $2xydy-y^2dx=0$

$2xydy-y^2dx=0$ $\frac{2xydy-y^2dx}{x^2}=0$ $d\left(\frac{y^2}{x}\right)=0$ On integration, we get $\frac{y^2}{x}=C$ $\therefore y^2=Cx$

### $(x+y)dy+(y-x)dx=0$

$(x+y)dy+(y-x)dx=0$ $xdy+ydy+ydx-xdx=0$ $(xdy+ydx)+ydy-xdx=0$ $d(xy)+d\left(\frac{1}{2}y^2\right)-d\left(\frac{1}{2}x^2\right)=0$ Integrating both sides, we get, $xy+\frac{1}{2}y^2-\frac{1}{2}x^2=C$ $\therefore 2xy+y^2-x^2=C$

### $xdy+(x+1)ydx=0$

$xdy+(x+1)ydx=0$ $xdy+xydx+ydx=0$ $\frac{dy}{y}+dx+\frac{dx}{x}=0$ $d(\log y)+dx+d(\log x)=0$ $d(\log xy+x)=0$ Integrating both sides, we get, $\log xy+x=C$

### $(x^2-ay)dx-(ax-y^2)dy=0$

$(x^2-ay)dx-(ax-y^2)dy=0$ $x^2dx-aydx-axdy+y^2dy=0$ $x^2dx-a(ydx+xdy)+y^2dy=0$ $d\left(\frac{1}{3}x^3\right)-ad(xy)+d\left(\frac{1}{3}y^3\right)=0$ $d\left(\frac{x^3}{3}-axy+\frac{y^3}{3}\right)=0$ On integration, $\frac{x^3}{3}-axy+\frac{y^3}{3}=0$ $\therefore x^3-3axy+y^3=C$

### $\sin x\cos xdx+\sin y\cos ydy=0$

$\sin x\cos xdx+\sin y\cos ydy=0$ $2\sin x\cos xdx+2\sin y\cos ydy=0$ $d(\sin^2x)+d(\sin^2y)=0$ $d(\sin^2x+\sin^2y)=0$ On integration, $\sin^2x+\sin^2y=C$

### $\frac{dy}{dx}=\frac{\cos^2y}{\sin^2y}$

$\frac{dy}{dx}=\frac{\cos^2y}{\sin^2y}$ $\frac{dy}{\cos^2y}=\frac{dx}{\sin^2x}$ $\sec^2ydy=\operatorname{cosec}^2xdx$ $d\tan y=-d\cot x$ $d(\tan y+\cot x)=0$ On integration, $\tan y+\cot x=C$

### $(x+2y-3)dy-(2x-y+1)dx=0$

$(x+2y-3)dy-(2x-y+1)dx=0$ $xdy+2ydy-3dy-2xdx+ydx-dx=0$ $(xdy+ydx)+2ydy-3dy-2xdx-dx=0$ $d(xy)+2d\left(\frac{y^2}{2}\right)-3dy-2d\left(\frac{x^2}{2}\right)-dx=0$ $d[xy+y^2-3y-x^2-x]=0$ On integration, $xy+y^2-3y-x^2-x=C$ $\therefore xy+y^2-x^2-3y-x=C$

### $\frac{dy}{dx}=\frac{y-x+1}{y-x+5}$

$\frac{dy}{dx}=\frac{y-x+1}{y-x+5}$ $(y-x+5)dy=(y-x+1)dx$ $ydy-xdy+5dy=ydx-xdx+dx$ $(ydx+xdy)-xdx+dx-ydy-5dy=0$ $d(xy)-d\left(\frac{x^2}{2}\right)+dx-d\left(\frac{y^2}{2}\right)-5dy=0$ $d\left[xy-\frac{x^2}{2}+x-\frac{y^2}{2}-5y\right]=0$ On integration, $xy-\frac{x^2}{2}+x-\frac{y^2}{2}-5y=C$ $\therefore 2xy-x^2+2x-y^2-10y=C$

### $(x^2+xy^2)dx+(x^2y+y^2)dy=0$

$(x^2+xy^2)dx+(x^2y+y^2)dy=0$ $x^2dx+xy^2dx+x^2ydy+y^2dy=0$ $d\left(\frac{x^3}{3}\right)+d\left(\frac{x^2y^2}{2}\right)+d\left(\frac{y^3}{3}\right)=0$ $d\left(\frac{x^3}{3}+\frac{x^2y^2}{2}+\frac{y^3}{3}\right)=0$ On integration, $\frac{x^3}{3}+\frac{x^2y^2}{2}+\frac{y^3}{3}=C$ $\therefore 2x^3+2y^3+3x^2y^2=C$

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