Differential Equation
Differential Equation
Differential Equation are two types
- Ordinary differential Equation(ODE)
- 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`
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