Question

Solve y = 2px – p² :

• A. $\frac{2}{3}$p + cp^–1
• B. $\frac{2}{3}$p + cp^–2
• C. $\frac{2}{3}$p + cp^–3
• D. None of these

Answer 1

Answer: (B)

$\frac{2}{3}$p + cp^–2

Answer 2

Differentiating the given equation with respect to x, we get ;

$\frac{dy}{dx}$ = 2p + 2x $\frac{dp}{dx}$ – 2p$\frac{dp}{dx}$

or 2 (x – p) $\frac{dp}{dx}$+ p = 0

or $\frac{dx}{dp}$ + $\frac{2}{p}x = 2$

It is a linear equation in x and p.

I.F. = $e^{\int_{}^{}{\frac{2}{p}dx}}$ = e² log p = p²

\ The solution is;

xp² = $\int_{}^{}p^{2}$ . 2 dp = $\frac{2}{3}$ p³ + c

Thus, the solution of the given equation is ;

x = $\frac{2}{3}$ p + cp^–2 where p is parameter