First slide

Methods of solving first order first degree differential equations

Question

$The solution of \frac{d y}{d x} = \frac{x^{2} + y^{2} + 1}{2 x y} satisfying y (1) = 1 is given by$

Moderate

A

a system of parabola
B

a system of circles
C

$y^{2} =x(1+x)-1$
D

${(x-2)}^{2} {+(y-3)}^{2} =5$

Solution

$\frac{d y}{d x} = \frac{x^{2} + y^{2} + 1}{2 x y} 2 x y d y = (x^{2} + y^{2} + 1) d x$
$2 x y d y - y^{2} d x = (x^{2} + 1) d x$
$\int \frac{{2xy}^{dy} {-y}^{2} dx}{x^{2}} = \int (1 + \frac{1}{x^{2}}) dx$
$\begin{array}{l} \to \int d (y^{2} / x) = x - \frac{1}{x} + c \\ \to \frac{y^{2}}{x} = x - \frac{1}{x} + c \\ \to y^{2} = (x^{2} - 1) + cx pass e s t h r o u g h (1, 1) \\ 1 = c \\ y^{2} = x^{2} - 1 + x \end{array}$

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