(d^2)/(dx^2)((y)(acos(2x)+bsin(2x)))

解

\frac{d ^{2}}{d x ^{2}} ((y) (a cos (2 x) + b sin (2 x)))

\frac{d ^{2} y}{d x ^{2}} (a cos (2 x) + b sin (2 x)) + 2 \frac{d y}{d x} (- 2 a sin (2 x) + 2 b cos (2 x)) + (- 4 a cos (2 x) - 4 b sin (2 x)) y

解答ステップ

\frac{d ^{2}}{d x ^{2}} ((y) (a cos (2 x) + b sin (2 x)))

\frac{d}{d x} (y (a cos (2 x) + b sin (2 x))) = \frac{d y}{d x} (a cos (2 x) + b sin (2 x)) + (- 2 a sin (2 x) + b cos (2 x) \cdot 2) y

= \frac{d}{d x} (\frac{d y}{d x} (a cos (2 x) + b sin (2 x)) + (- 2 a sin (2 x) + b cos (2 x) \cdot 2) y)

\frac{d}{d x} (\frac{d y}{d x} (a cos (2 x) + b sin (2 x)) + (- 2 a sin (2 x) + b cos (2 x) \cdot 2) y) = \frac{d ^{2} y}{d x ^{2}} (a cos (2 x) + b sin (2 x)) + 2 \frac{d y}{d x} (- 2 a sin (2 x) + 2 b cos (2 x)) + (- 4 a cos (2 x) - 4 b sin (2 x)) y

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