First slide

Methods of solving first order first degree differential equations

Question

$The solution of the D.E (2 y + x y^{3}) d x + (x + x^{2} y^{2}) d y = 0 is$

Moderate

A

$x^{2} y+ \frac{x^{3} y^{3}}{3} =C$
B

${xy}^{2} + \frac{x^{3} y^{3}}{3} =C$
C

$x^{2} y+ \frac{x^{4} y^{4}}{4} =C$
D

$x^{2} y- \frac{x^{3} y^{3}}{3} =C$

Solution

$\begin{array}{l} (2 ydx + xdy) + ({xy}^{3} dx + x^{2} y^{2} dy) = 0 \\ Multiplying by x \\ (2 xydx + x^{2} dy) + (x^{2} y^{3} dx + x^{3} y^{2} dy) = 0 \end{array}$
$(2 xy + x^{2} \frac{dy}{dx}) + (x^{2} y^{3} + x^{3} y^{2} \frac{dy}{dx}) = 0$
$\begin{array}{l} \to d (x^{2} y) + \frac{1}{3} d (x^{3} y^{3}) = 0 \\ \to \int d (x^{2} y) + \frac{1}{3} \int d (x^{3} y^{3}) = 0 \\ \to x^{2} y + \frac{x^{3} y^{3}}{3} = C \end{array}$

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