The Clairaut Differential Equation With Scientific Point Of View
Clairaut's Differential Equations : The equation y= px+ f(p) . . . . . (1) of first degree in x and y is called the ' Claitaut's equation ' after the name of Alex Claude CLAIRAUT(1713_1765). Differentiating w.r.t x, bearing in mind that p= dy/ dx, we obtain p= xdp/dx+ p+ f(p)dp/ dx or [x+f(p)]dp/dx=0. Equating each factor to zero, we get dp/ dx=0. .....(2) and x+ f(p)=0. .......(3) Integrating (2) we obtain p=c ( a constant). Putting this value of p into (1), we find its complete integral y= cx+ f(c). ............(4) which represents geometrically, a family of straight lines. Elimination of p between (3) and (1) leads to a singular solution. Example : Solve y= px+ a/p Solution. Differentiating w.r.t x, we get p= p+x dp/dx- a/p^2 dp/dx or (x- a/p^2)dp/dx=0 Taking dp/dx=0, we get p=c Substituting this value of p in the give