Question

Find the complete integral of p - 3 x² = q² - y

Answer 1

The given equation is a separable equation of the form $f(x, p)=g(y, q)$.

The complete solution will be, $z = \int pdx + \int qdy + c$

Let $f(x, p)=g(y, q) =a$. Solving for $f(x, p)=a~ \&~ g(y, q) =a$, we get

$\begin{aligned} p-3x^2 &= a&; & ~q^2 -y = a\\ p&=3x^2 + a&; & ~q^2=y+a\\ p&=3x^2 + a&; & ~q=\sqrt{y+a}\\ \end{aligned}$

The complete solution is,

$\begin{aligned} z &= \int pdx + \int qdy + c\\ z &= \int (3x^2 + a)dx + \int (\sqrt{y+a})dy + c\\ &= x^{3} + ax + \frac{2}{3}(y+a)^{\frac{3}{2}} + c \end{aligned}$