First slide

Methods of solving first order first degree differential equations

Question

$The solution of the differential equation \frac{dy}{dx} - \frac{y + 3 x}{\log_{e} (y + 3 x)} + 3 = 0 is:$ (where C is a constant of integration.)

Moderate

A

$x - \log_{e} (y + 3 x) = C$
B

$x - 2 \log_{e} (y + 3 x) = C$
C

$x - \frac{1}{2} {(\log_{e} (y + 3 x))}^{2} = C$
D

$y + 3 x - \frac{1}{2} {(\log_{e} x)}^{2} = C$

Solution

$\begin{array}{l} \frac{dy}{dx} - \frac{y + 3 x}{\ln (y + 3 x)} + 3 = 0 \\ dy + 3 dx = \frac{(y + 3 x)}{\ln (y + 3 x)} dx \\ \frac{\ln (y + 3 x)}{y + 3 x} (d (y + 3 x)) = dx i n t e g r a t e b o t h s i d e s \\ \frac{\ln^{2} (y + 3 x)}{2} = x + c (let y + 3 x = t then the equation becomes \int \frac{\ln t}{t} dt = \frac{{(\ln t)}^{2}}{2}) \\ or x - \frac{1}{2} \ln^{2} (y + 3 x) = C \end{array}$

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