First slide

Methods of solving first order first degree differential equations

Question

$The solution of the differential equation 3 {xy}^{1} - 3 y + {(x^{2} - y^{2})}^{1 / 2} = 0 satisfying the condition y(1)=1 is$

Easy

A

${3cos}^{-1} (y/x)=ln |x|$
B

$3cos(y/x)=ln |x|$
C

${3cos}^{-1} (y/x)=2log |x|$
D

${3sin}^{-1} (y/x)=ln |x|$

Solution

$\begin{array}{l} - \frac{dy}{dx} = \frac{3 y - \sqrt{x^{2} - y^{2}}}{3 x}, put y = vx \\ \frac{d y}{d x} = v + \frac{x d v}{d x} \\ \int \frac{3}{\sqrt{1 - v^{2}}} dv = - \int \frac{dx}{x} \Rightarrow 3 \cos^{- 1} (y / x) = \log | x | + c \\ \underline{\underline{Pass} (1, 1)} \Rightarrow c = 0 \end{array}$
${3cos}^{-1} (y/x)=ln |x|$

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