The general solution of the differential equa | vedclass

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Question

(B) Given differential equation is $\frac{y dx - x dy}{y} = 0$.Divide both sides by $xy$ (assuming $x 
eq 0, y 
eq 0$):$\frac{y dx - x dy}{xy} = 0$T

Vedclass · Accepted Answer

$y = C x$

Vedclass · Answer

(B) Given differential equation is $\frac{y dx - x dy}{y} = 0$.Divide both sides by $xy$ (assuming $x 
eq 0, y 
eq 0$):$\frac{y dx - x dy}{xy} = 0$This can be rewritten as:$\frac{dx}{x} - \frac{dy}{y} = 0$Integrating both sides with respect to their variables:$\int \frac{1}{x} dx - \int \frac{1}{y} dy = \int 0 dx$$\log |x| - \log |y| = \log |C|$Using the logarithmic property $\log a - \log b = \log(\frac{a}{b})$:$\log \left| \frac{x}{y} ight| = \log |C|$Taking the exponential of both sides:$\frac{x}{y} = C$Rearranging for $y$:$y = \frac{1}{C} x$Since $\frac{1}{C}$ is an arbitrary constant,we can write it as $C_1$:$y = C_1 x$Thus,the correct option is $B$.

The general solution of the differential equation $\frac{y dx - x dy}{y} = 0$ is

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