Question

solve. xp2-2yp+x+2y=0​

Answer 1

Step-by-step explanation:

In order to use Clairaut's technique, we write (a) as

(p2+1)x+2(1−p)y=0⟹y=p2+12(p−1)x

Now, deriving with respect to x,

dydx=p=ddx{p2+12(p−1)}x+p2+12(p−1),

or

p−p2+12(p−1)=p2−2p−12(p−1)2p′(x)x.

Simplifying,

p′(x)p−1=1x,

which can be integrated, yielding to p(x)=1+c1x, and then y(x)=c12x2+x+c2. Now, in order to determine the extra constant, we substitute into (a), and then

y(x)=c12x2+x+1c1.