Answer to Question #152055 in Differential Equations for Aritra Chatterjee

Question

Solution of : xp^2 - (x-a)^2 = 0

Answer 1

$xp^2-(x-a)^2=0$

we have difference of squares

$(\sqrt{x}p-(x-a))(\sqrt{x}p+(x-a))=0$

Two equations and two solutions:

first equation:

$\sqrt{x}p-x+a=0$

$p=\frac{x-a}{\sqrt{x}}=\sqrt{x}-\frac{a}{\sqrt{x}}$

first solution:

$z=\frac{2}{3}x^{\frac{3}{2}}-2a\sqrt{x}+C$

second equation:

$\sqrt{x}p+x-a=0$

$p=-\frac{x-a}{\sqrt{x}}=-\sqrt{x}+\frac{a}{\sqrt{x}}$

second solution:

$z=-\frac{2}{3}x^{\frac{3}{2}}+2a\sqrt{x}+C$