Answer to Question #310735 in Differential Equations for Asha

Question #310735

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


1
Expert's answer
2022-03-14T17:28:37-0400

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


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


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

p3x2=a; q2y=ap=3x2+a; q2=y+ap=3x2+a; q=y+a\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,


z=pdx+qdy+cz=(3x2+a)dx+(y+a)dy+c=x3+ax+23(y+a)32+c\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}


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