First slide

Methods of solving first order first degree differential equations

Question

$A curve passes through the point (1, π / 6) . Let the slope of the curve at each$ $point (x, y) be \frac{y}{x} + \sec (y / x), x > 0 then the equation of the curve is$

Moderate

A

$\sin (y/x) = logx + \frac{1}{2}$
B

$cos e c (y/x) = logx + 2$
C

$s e c (2y/x) = logx + 2$
D

$cos (2y/x) = logx + \frac{1}{2}$

Solution

$\begin{array}{l} \frac{dy}{dx} = \frac{y}{x} + \sec (y / x) put y = v x \\ v + x \frac{d v}{d x} = v + s e c v \\ \to \frac{dv}{secv} = \frac{dx}{x} \to \int cosvdv = \int \frac{dx}{x} \\ \to \sin (y / x) = \ln x + c \\ The curve passes through (1, π / 6) is \end{array}$
$\begin{array}{l} \to \frac{1}{2} = c \\ \to \sin (y / x) = \ln x + \frac{1}{2} \end{array}$

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