x^3 y′′′ − 3x^2 y′′+6xy′ −6y=x^4 ln(x),x>0


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Question Number 78878 by M±th+et£s last updated on 21/Jan/20

$x^{3} y^{‴} - 3 x^{2} y^{″} + 6 {x y}^{'} - 6 y = x^{4} l n (x), x > 0$
	Answered by mind is power last updated on 21/Jan/20
	$y=x^a Solution of homgenius⇒ x^3 y′′′−3x^2 y′′+6xy′−6y (a(a−1)(a−2)−3a(a−1)+6a−6)y(x)=0 ⇒ (a−1)(a(a−2)−3a+6)=0 (a−1)(a^2 −5a+6)=0 ⇒(a−1)(a−3)(a−2) ⇒a∈{1,2,3} y_1 =x,y_2 =x^2 ,y_3 =x^3 ,y_1 ,y_2 ,y_3 3 independent Solution ⇒ Y=ay_1 +by_2 +cy_3 Solution of Homgenius Equation let y=g(x)x particular Solution y′=xg′+g y′′=2g′+xg′′ y′′′=3g′′+xg′′′ ⇒x^3 (3g′′+xg′′′)−3x^2 (2g′+xg′′)+6x(xg′+g)−6xg(x)=x^4 ln(x) ⇒x^4 g′′′=x^4 ln(x)⇒g′′′(x)=ln(x) ⇒g′′=xln(x)−x g′(x)=((x^2 ln(x))/2)−(x^2 /4)−(x^2 /2)=((x^2 ln(x))/2)−((3x^2 )/4) g(x)=((x^3 ln(x))/6)−(x^3 /(18))−(x^3 /4)=((x^3 ln(x))/6)−((11x^3 )/(36)) y(x)=ax+bx^2 +cx^3 +(x^4 /6)(ln(x)−((11)/6))$
	$y = x^{a} S o l u t i o n o f h o m g e n i u s \Rightarrow$ $x^{3} y^{‴} - 3 x^{2} y^{″} + 6 {x y}^{'} - 6 y$ $(a (a - 1) (a - 2) - 3 a (a - 1) + 6 a - 6) y (x) = 0$ $\Rightarrow$ $(a - 1) (a (a - 2) - 3 a + 6) = 0$ $(a - 1) (a^{2} - 5 a + 6) = 0$ $\Rightarrow (a - 1) (a - 3) (a - 2)$ $\Rightarrow a \in {1, 2, 3}$ $y_{1} = x, y_{2} = x^{2}, y_{3} = x^{3}, y_{1}, y_{2}, y_{3} 3 i n d e p e n d e n t S o l u t i o n \Rightarrow$ $Y = {a y}_{1} + {b y}_{2} + {c y}_{3} S o l u t i o n o f H o m g e n i u s E q u a t i o n$ $l e t y = g (x) x p a r t i c u l a r S o l u t i o n$ $y^{'} = {x g}^{'} + g$ $y^{″} = 2 g^{'} + {x g}^{″}$ $y^{‴} = 3 g^{″} + {x g}^{‴}$ $\Rightarrow x^{3} (3 g^{″} + {x g}^{‴}) - 3 x^{2} (2 g^{'} + {x g}^{″}) + 6 x ({x g}^{'} + g) - 6 x g (x) = x^{4} l n (x)$ $\Rightarrow x^{4} g^{‴} = x^{4} l n (x) \Rightarrow g^{‴} (x) = l n (x)$ $\Rightarrow g^{″} = x l n (x) - x$ $g^{'} (x) = \frac{x^{2} l n (x)}{2} - \frac{x^{2}}{4} - \frac{x^{2}}{2} = \frac{x^{2} l n (x)}{2} - \frac{3 x^{2}}{4}$ $g (x) = \frac{x^{3} l n (x)}{6} - \frac{x^{3}}{18} - \frac{x^{3}}{4} = \frac{x^{3} l n (x)}{6} - \frac{11 x^{3}}{36}$ $y (x) = a x + {b x}^{2} + {c x}^{3} + \frac{x^{4}}{6} (l n (x) - \frac{11}{6})$
	Commented byM±th+et£s last updated on 21/Jan/20

	$g o d b l e s s y o u s i r$
	Commented bymind is power last updated on 21/Jan/20

	$T h a n x y o u T o s i r$
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