Differential equations are solved by reducing them to the exact differential of an expression in \( x \) and \( y \) i.e., they are reduced to the form \( d(f(x, y))=0 \)
Differential equation \( y=p x+f(p) \), wherep \( p=\frac{d y}{d x} \), is known as Clairouts Equation.To solve equation (1),differentiate it with respect to \( x \), which gives either
\[
\begin{array}{l}
\frac{d p}{d x}=0 \Rightarrow p=c \\
\text { or } x+f^{\prime}(p)=0
\end{array}
\]
Note:
(i) If \( p \) is eliminated between equations (1) and (2), the solution obtained is a general solution of equation.(1).
(ii) If \( p \) is eliminated between equation (1) and (3) then solution obtained does not contain any arbitrary constant and is not particu1ar solution of equation (1). This solution is called singular solution of equation (1).
The number of solution of the equation \( f(x)=-1 \) and the singular Solution of the equationy \( y=x \frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{2} \) is
(a) 1
(b) 2
(c) 4
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