Question

Find the general and singular solution of xp^2-yp+a=0

Answer 1

Find the general and singular solution of

xp²-yp+a=0

Make y the subject of the formula

y=xp+a/p (Since this is a clairaut’s equation)

It general solution will be given by

put p=c

Then y=cx+ a/c

The general solution is   y=cx+ a/c

Differentiate with respect to c

0=x(1) - a/c² 

c²=a/x

c=$\sqrt{a/c}$  

Put c into the general equation 

y=x$\sqrt{a/c}$ +a$\sqrt{x/a}$

The singular solution is y=x$\sqrt{a/c}$ +a$\sqrt{x/a}$