The Clairaut Differential Equation With Scientific Point Of View
Clairaut's Differential Equations :
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:
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 given equation we obtain the complete integral
y= cx+ a/c
Taking x-a/p^2=0 and eliminating p between this and the given equation we obtain the singular solution
y^2= 4ax
ABOUT SCIENTIST
Alexis Claude Clairaut (French: [klɛʁo]; 13 May 1713 – 17 May 1765) was a French mathematician, astronomer, and geophysicist. He was a prominent Newtonian whose work helped to establish the validity of the principles and results that Sir Isaac Newton had outlined in the Principia of 1687. Clairaut was one of the key figures in the expedition to Lapland that helped to confirm Newton's theory for the figure of the Earth. In that context, Clairaut worked out a mathematical result now known as "Clairaut's theorem". He also tackled the gravitational three-body problem, being the first to obtain a satisfactory result for the apsidal precessionof the Moon's orbit. In mathematics he is also credited with Clairaut's equation and Clairaut's relation.
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