The solution of d y d x = y x + t a n y x is Options: A . &nbsp y s i n ( y x ) = c x B . &nbsp y s i n ( y x ) = c y C . &nbsp s i n ( x y ) = c x D . &nbsp s i n ( y x ) = c x

Answer: Option D : D Put y = v x . T h e n d y d x = v + x d v d x Given equation is d y d x = v x + t a n y x ⇒ v + x t a n v ⇒ c o t v d v = d x x ⇒ l o g s i n v = l o g x + l o g c ⇒ s i n v = c x ⇒ s i n ( y x = c x )

The solution of dydx=yx+tanyx is

Discussion Forum : Differential Equations

Question -

The solution of $\frac{d y}{d x} = \frac{y}{x} + t a n \frac{y}{x}$ is

Options:

A .

y s i n (\frac{y}{x}) = c x

B .

y s i n (\frac{y}{x}) = c y

C .

s i n (\frac{x}{y}) = c x

D .

s i n (\frac{y}{x}) = c x

Answer: Option D
:
D
Put

y = v x . T h e n \frac{d y}{d x} = v + x \frac{d v}{d x}

Given equation is

\frac{d y}{d x} = \frac{v}{x} + t a n \frac{y}{x} \Rightarrow v + x t a n v \Rightarrow c o t v d v = \frac{d x}{x} \Rightarrow l o g s i n v = l o g x + l o g c

\Rightarrow s i n v = c x \Rightarrow s i n (\frac{y}{x} = c x)

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