(d^3)/(dx^3)((y)(2x^2-1/(x^3)))

Lösung

\frac{d ^{3}}{d x ^{3}} ((y) (2 x^{2} - \frac{1}{x ^{3}}))

2 x^{2} \frac{d ^{3} y}{d x ^{3}} + 12 x \frac{d ^{2} y}{d x ^{2}} - \frac{\frac{d ^{3} y}{d x ^{3}}}{x ^{3}} + \frac{9 \frac{d ^{2} y}{d x ^{2}}}{x ^{4}} + 12 \frac{d y}{d x} - \frac{36 \frac{d y}{d x}}{x ^{5}} + \frac{60 y}{x ^{6}}

Schritte zur Lösung

\frac{d ^{3}}{d x ^{3}} ((y) (2 x^{2} - \frac{1}{x ^{3}}))

\frac{d}{d x} (y (2 x^{2} - \frac{1}{x ^{3}})) = \frac{d y}{d x} (2 x^{2} - \frac{1}{x ^{3}}) + (4 x + \frac{3}{x ^{4}}) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (2 x^{2} - \frac{1}{x ^{3}}) + (4 x + \frac{3}{x ^{4}}) y)

\frac{d}{d x} (\frac{d y}{d x} (2 x^{2} - \frac{1}{x ^{3}}) + (4 x + \frac{3}{x ^{4}}) y) = \frac{d ^{2} y}{d x ^{2}} (2 x^{2} - \frac{1}{x ^{3}}) + 2 (4 x + \frac{3}{x ^{4}}) \frac{d y}{d x} + (4 - \frac{12}{x ^{5}}) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x^{2} - \frac{1}{x ^{3}}) + 2 (4 x + \frac{3}{x ^{4}}) \frac{d y}{d x} + (4 - \frac{12}{x ^{5}}) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (2 x^{2} - \frac{1}{x ^{3}}) + 2 (4 x + \frac{3}{x ^{4}}) \frac{d y}{d x} + (4 - \frac{12}{x ^{5}}) y) = 2 x^{2} \frac{d ^{3} y}{d x ^{3}} + 12 x \frac{d ^{2} y}{d x ^{2}} - \frac{\frac{d ^{3} y}{d x ^{3}}}{x ^{3}} + \frac{9 \frac{d ^{2} y}{d x ^{2}}}{x ^{4}} + 12 \frac{d y}{d x} - \frac{36 \frac{d y}{d x}}{x ^{5}} + \frac{60 y}{x ^{6}}

= 2 x^{2} \frac{d ^{3} y}{d x ^{3}} + 12 x \frac{d ^{2} y}{d x ^{2}} - \frac{\frac{d ^{3} y}{d x ^{3}}}{x ^{3}} + \frac{9 \frac{d ^{2} y}{d x ^{2}}}{x ^{4}} + 12 \frac{d y}{d x} - \frac{36 \frac{d y}{d x}}{x ^{5}} + \frac{60 y}{x ^{6}}

e x p an d \frac{4 e ^{- 2 s}}{( s - 1 ) ( s + 4 ) ^{2}}

e x p an d \frac{x + 2}{2 ( x - 2 ) ( x + 1 )}

e x p an d P (6, 5) y Q (2, - 3)

e x p an d F (3, - \frac{3}{4})

e x p an d \frac{( x ^{2} - 2 x + 3 )}{( x - 3 ) ( x ^{2} + 9 )}