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 books.google.com StepbyStep Solutions to Problems in this Chapter, "Clairaut's Equation" •
PROBLEM 91 Solve the differential equation (xy1  y)3  (y')a  1 J Solution;
Since x and y occur only in the combination (xy'  y) this is a Clairaut equation.
The general ... 

 books.google.com Clairaut A class of equations with interesting properties was found by Clairaut (
see Clairaut (1734), Probleme 111). He was motivated by the movement of a
rectangular wedge (see Fig. 2.5), which led him to differential equations of the
form ... 

 books.google.com The discrete analog of Clairaut's differential equation y = tj/ + /(?/) appears as (3.1
.1) u(k) = kAti(k) + f(Au(k)), keVi where / is some nonlinear function. In the al>ove
difference equation let v(k) = Au(k), so that (3.1.2) u(k) = kv(k) + f(v(k)) and v(k) m
... 

 books.google.com 446 Clairaut's Difference Equation Clark's Triangle TION, CONTINUOUS
FUNCTION, DIFFERENTIAL EQUATION, REGULARITY (PDE) References
Cartan, H. Cours de calcul differentiel. Paris: Hermann, 1977. Krantz, S. G. "
Continuously ... 

 books.google.com Solve xp2 — yp + a = 0. Sol. The given differential equation can be written as yp
= xp2 + a or y = xp + a/p, which is in Clairaut's form. Hence replacing p by the
arbitrary constant c in the given differential equation, the required solution isxc2
—yc ... 

 books.google.com An equation of the form () y xpfp=+, (5) where () f p does not contain x or y
explicitly, is called Clairaut's equation. Clearly this equation can be solved using
the technique of section 4.2. (Equations solvable for y). But, because the solution
of this ... 

 books.google.com Thus, the solution of Clairaut's equation is obtained by writing с forp in the
equation. Example 1. Solve (y px)(p  1) =p. Sol. We have (y  px)(p  1) = p ypx
= pl y=px P pl ...d) ...(2) (2) is of the form у = px + Др) with Др) = .. (2) is a
Clairaut's ... 

 books.google.com It is of Clairaut's form, hence its solution is y=cx+ c — 1 (ii) The given equation is
sin px cos y  cos px sin y = p or sin (px — y) = p or px — y = sin"1 p or y = px —
sin—1 p It is of Clairaut's form and hence its solution is y = we  sin1 c. (iii)
Please ... 

 books.google.com Case 8 Clairaut's Equation A differential equation of the fonn y = PX + f (p) is
known as Clairaut's equation, where f is an arbitrary function. Differentiating this
equation with respect to x, we obtain , d {x +f (11)} —p = 0 dx From which, we get
... 

 books.google.com Suppose we have the ordinary differential equation (xy1  yf  (y1)2 1=0. (50.4)
Because equation ... Clairaut's partial differential equation z = \^ x i '577 + '/ ( SxT'
•••' dx^) ;=i has the solution z = £]"=i aixi + f(ai>a2, □ □ □ ,o,n). See Kamke [3 ... 

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