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Monday, 17 February 2014

Singular Solutions


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    P(x,y)      on the envelope . They exists a curve of the family , say;


Which touches the envelope at    (x,y).   The values of    x,y,dy/dx    for the curve at     satisfies the given differential equations. Also the value of   x,y,dy/dx   at    for the envelope are the same as for the curve. Thus we see that the values of the   x,y,dy/dx   at every point of the envelope satisfy the given differential equation. Hence the equation of the envelope is a solution of differential equation.

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 .