Exact Form (Standard Form IV) | Differential Equations

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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