JEE Main & Advanced Mathematics Trivarna Dhvajah Exact Differential Equation

(1) Exact differential equation : If \[M\] and \[N\] are functions of \[x\] and \[y,\] the equation \[Mdx+Ndy=0\]  is called exact when there exists a function \[f(x,\,\,y)\] of \[x\] and \[y\] such that

\[d[f(x,\,\,y)=Mdx+Ndy\] i.e., \[\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy=Mdx+Ndy\]

where \[\frac{\partial f}{\partial x}=\]Partial derivative of \[f(x,\,y)\] with respect to \[x\] (keeping y constant) . \[\frac{\partial f}{\partial y}=\]Partial derivative of \[f(x,\,\,y)\] with respect to \[y\] (treating \[x\]as constant)

The necessary and sufficient condition for the differential condition \[Mdx+Ndy=0\] to be exact is \[\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\].

An exact differential equation can always be derived from its general solution directly by differentiation without any subsequent multiplication, elimination etc.

(2) Integrating factor : If an equation of the form \[Mdx+Ndy=0\] is not exact, it can always be made exact by multiplying by some function of \[x\] and \[y\]. Such a multiplier is called an integrating factor.

(3) Working rule for solving an exact differential equation :

Step (i) : Compare the given equation with \[Mdx+Ndy=0\] and find out \[M\] and \[N\]. Then find out \[\frac{\partial M}{\partial y}\] and \[\frac{\partial N}{\partial x}\]. If \[\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}\], the given equation is exact.

Step (ii) : Integrate \[M\] with respect to \[x\] treating \[y\] as a constant.

Step (iii) : Integrate \[N\] with respect to \[y\] treating \[x\] as constant and omit those terms which have been already obtained by integrating \[M\].

Step (iv) : On adding the terms obtained in steps (ii) and (iii) and equating to an arbitrary constant, we get the required solution.

In other words, solution of an exact differential equation is

\[\underset{\begin{smallmatrix} \text{Regarding }y\text{ } \\  \text{as constant}  \end{smallmatrix}}{\mathop{\int_{{}}^{{}}{Mdx}}}\,+\underset{\begin{smallmatrix} \text{Only those terms } \\  \text{not containing }x \end{smallmatrix}}{\mathop{\int_{{}}^{{}}{Ndy}}}\,=c\]