Question

10. Reduce the differential equation y = 2 px + y2p to Clairaut's form by putting
= v and hence find its general and singular solutions.
and test for singular solutions 3 - 4pxy + 8 y2 = 0.
yt​

Answer 1

Given the differential equation,

 y=px+p−p2.......(1)

Now, differentiating both sides with respect to x we get,

p=p+(x+1−2p)dxdp [ Since dxdy=p]

or, (x+1−2p)dxdp=0

or, x+1=2p and dxdp=0.

or, p=2x+1 gives particular solution and dxdp=0 leads to general solution.

Now putting p=2x+1 in (1) we get,

y=2(x+1)2−4(x+1)2

or, y=4(x+1)2

or, (x+1)2=4y.