In addition to the General Solutions and Particular Solutions, obtained by giving particular values to the arbitrary constant in general solution a differential equation , may also posses other solutions . The solutions of differential equations , other then the general and particular , are known as Singular Solutions . In this connection , we have the following result .

Whenever the family of curves

**f(x,y,c)=0 --------------(1)**represented by the differential equation

**phai(x,y,dy/dx)=0 -------(2)**posses an envelope , the equation of the envelope is the singular solution of the differential equation

*.*

**(2)**Suppose that the family of curves possesses an envelope. Take any point

*on the envelope . They exists a curve of the family , say;*

**P(x,y)**

**f(x,y,c')=0**Which touches the envelope at

*The values of*

**(x,y).***for the curve at*

**x,y,dy/dx***satisfies the given differential equations. Also the value of*

**P***at*

**x,y,dy/dx***for the envelope are the same as for the curve. Thus we see that the values of the*

**P***at every point of the envelope satisfy the given differential equation. Hence the equation of the envelope is a solution of differential equation.*

**x,y,dy/dx**This solution does not contain any arbitrary constant and in general can not be obtained from the general solution by giving particular values to the arbitrary constant .

## No comments:

## Post a Comment