Answer to Question #153732 in Differential Equations for uday

Question

y²p² -3xy+y=0

Answer 1

WE HAVE TO SOLVE THE D.E. GIVEN ABOVE .

SOLUTION:-

$y^2p^2-3xy+y=0\\\implies\sqrt y p-\sqrt {(3x-1)}=0\\\implies\sqrt y\frac{dy}{dx}-\sqrt{(3x-1)}=0\\\implies\int\sqrt ydy-\int\sqrt {(3x-1)}dx=0\\\implies\frac{2}{3}y^\frac{3}{2}-\frac{2}{9}(3x-1)^\frac{3}{2}=c'\\ [c' \; is \; a \;constant]\\ \implies3y^\frac{3}{2}-(3x-1)^\frac{3}{2}=c\\ [c=9c'/2]$

Hence the solution of the given equation:

$\boxed{\pmb{3y^\frac{3}{2}-(3x-1)^\frac{3}{2}=c}}$