x(dy/dx)−6(y/x)=7x^3 help me


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Question Number 125054 by micelle last updated on 08/Dec/20

$x \frac{d y}{d x} - 6 \frac{y}{x} = 7 x^{3}$ $h e l p m e$
	Answered by Dwaipayan Shikari last updated on 08/Dec/20

	$\frac{d y}{d x} - \frac{6 y}{x^{2}} = 7 x^{4}$ $I . F = e^{\int - \frac{6}{x^{2}}} = e^{\frac{6}{x}}$ ${y e}^{\frac{6}{x}} = 7 \int x^{2} e^{\frac{1}{x}} d x$ $y = 7 e^{- \frac{6}{x}} \underset{n = 0}{\sum^{\infty}} \frac{x^{1 - n}}{n! (1 - n)} + 7 {C e}^{- \frac{6}{x}}$
	Answered by TITA last updated on 08/Dec/20

	$\frac{d y}{d x} - 6 \frac{y}{x^{2}} = 7 x^{2} u s i n g i n t e r g r a t i n g f a c t o r$ $\emptyset = e^{\int \frac{- 6}{x^{2}} d x} = e^{\frac{6}{x}} h e n c e e^{\frac{6}{x}} \frac{d y}{d x} - 6 e^{\frac{6}{x}} \frac{y}{x^{2}} = 7 e^{\frac{6}{x}} x^{2}$ $\frac{d ({y e}^{\frac{6}{x}})}{d x} = 7 x^{2} \Rightarrow {y e}^{\frac{6}{x}} = 7 \int e^{\frac{6}{x}} x^{2} d x$
	Commented by TITA last updated on 08/Dec/20

	$p l e a s e c o n t i n u e f r o m t h e r e$
	Commented by benjo_mathlover last updated on 08/Dec/20

	$i t s h o u l d b e {y e}^{\frac{6}{x}} = \int 7 x^{2} e^{\frac{6}{x}} d x .$ $y o u r s o l u t i o n i s w r o n g$
	Commented by TITA last updated on 08/Dec/20

	$o k t h a n k s$
	Commented by mohammad17 last updated on 08/Dec/20

	$f a l s e s o l u t i o n$
	Answered by Bird last updated on 08/Dec/20

	$e \to y^{^{'}} - \frac{6}{x^{2}} y = 7 x^{2}$ $h \to \frac{y^{^{'}}}{y} = \frac{6}{x^{2}} \Rightarrow l n ∣ y ∣= - \frac{6}{x} + c \Rightarrow$ $y = k e^{- \frac{6}{x}} l a v r a n g e m e t h o d \to$ $y^{^{'}} = k^{^{'}} e^{- \frac{6}{x}} + k (\frac{6}{x^{2}}) e^{- \frac{6}{x}}$ $e \Rightarrow k^{^{'}} e^{- \frac{6}{x}} + \frac{6 k}{x^{2}} e^{- \frac{6}{x}} - \frac{6}{x^{2}} {k e}^{- \frac{6}{x}} = 7 x^{2}$ $\Rightarrow k^{^{'}} = 7 x^{2} e^{- \frac{6}{x}} \Rightarrow$ $k = \int 7 x^{2} e^{- \frac{6}{x}} d x + λ \Rightarrow$ $y (x) = (\int^{x} 7 t^{2} e^{- \frac{6}{t}} d t + λ) e^{- \frac{6}{x}}$
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