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

Solución

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

- π^{2} x^{2} cos (π x y) y \frac{d y}{d x}^{2} - 2 π^{2} x cos (π x y) y^{2} \frac{d y}{d x} - 2 π x sin (π x y) \frac{d y}{d x}^{2} - π x sin (π x y) y \frac{d ^{2} y}{d x ^{2}} - π^{2} cos (π x y) y^{3} - 4 π sin (π x y) y \frac{d y}{d x} + \frac{- 4 x ^{3} y \frac{d ^{2} y}{d x ^{2}} + 12 x ^{3} \frac{d y}{d x} ^{2} + x ^{3} cos ( π x y ) y ^{4} \frac{d ^{2} y}{d x ^{2}} - x ^{2} y ^{4} \frac{d ^{2} y}{d x ^{2}} + 2 x y ^{4} \frac{d y}{d x} - 2 y ^{5}}{x ^{3} y ^{4}}

Pasos de solución

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

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

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

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

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

e x p an d sin (x) (1 + tan (x)) + cos (x) (1 + cot (x))

e x p an d 18 \cdot \frac{7}{3}

s im pl i f y (1 + i 3)^{8}

e x p an d \frac{( 4 x ^{6} - 1 ) ^{2}}{16 x ^{6}}

e x p an d (- 1 + - 25 x^{2} - 10 x + 101)^{3}