IIT JAM Mathematics 2022 Short Note on Differential Equation.

Differential Equation

Differential Equation are two types

1.  Ordinary differential Equation(ODE)
2. Partial differential Equation(PDE)

Ordinary Differential Equation(ODE):

• Linear Differential Equation .
• Non Linear Differential Equation.           

Method Of Solving Differential Equations:

    Variable Separable Method:

`f\left(x\right)dx+g\left(y\right)dy\;=c` 

 Integrating both side, we get the solution ,

`\int f\left(x\right)dx+\int g\left(y\right)dy\;=c`

Exact Differential Equation:

If M and N are function of x and y , the differential equation Mdx+Ndy=0 is exact if there exist a function f(x,y) such that d[f(x,y)]=Mdx+Ndy .

 

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

Equation Reducible to Exact::

   

Rule-I

If The equation Mdx+Ndy=0 is homogeneous then `\frac1{Mx+Ny}` is an integrating factor provider Mx+Ny not equal to 0.

Rule-II

If The equation Mdx+Ndy=0 is The form `f_1(xy)ydx+f_2(xy)xdy=0`  Then `\frac1{Mx+Ny}`is an integrating factor provider Mx+Ny not equal to 0.

Rule-III

If `\frac1N\left(\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}\right)=f(x)\;then\;e^{\int f(x)dx}` is an I.F of the equation Mdx+Ndy=0.

Or, 

If `\frac1M\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)=f(y)\;then\;e^{\int f(y)dy}` is an I.F of the equation Mdx+Ndy=0.

Rule-IV

`x^hy^k` (Mdx+Ndy)  then find the value h and k then I.F=`x^hy^k`