(d^2)/(dx^2)((y)(e^{x^2}+sin(3x+1)))

解

\frac{d ^{2}}{d x ^{2}} ((y) (e^{x^{2}} + sin (3 x + 1)))

\frac{d ^{2} y}{d x ^{2}} (e^{x^{2}} + sin (3 x + 1)) + 2 \frac{d y}{d x} (2 e^{x^{2}} x + 3 cos (3 x + 1)) + (2 (2 e^{x^{2}} x^{2} + e^{x^{2}}) - 9 sin (3 x + 1)) y

解答ステップ

\frac{d ^{2}}{d x ^{2}} ((y) (e^{x^{2}} + sin (3 x + 1)))

\frac{d}{d x} (y (e^{x^{2}} + sin (3 x + 1))) = \frac{d y}{d x} (e^{x^{2}} + sin (3 x + 1)) + (e^{x^{2}} \cdot 2 x + cos (3 x + 1) \cdot 3) y

= \frac{d}{d x} (\frac{d y}{d x} (e^{x^{2}} + sin (3 x + 1)) + (e^{x^{2}} \cdot 2 x + cos (3 x + 1) \cdot 3) y)

\frac{d}{d x} (\frac{d y}{d x} (e^{x^{2}} + sin (3 x + 1)) + (e^{x^{2}} \cdot 2 x + cos (3 x + 1) \cdot 3) y) = \frac{d ^{2} y}{d x ^{2}} (e^{x^{2}} + sin (3 x + 1)) + 2 \frac{d y}{d x} (2 e^{x^{2}} x + 3 cos (3 x + 1)) + (2 (2 e^{x^{2}} x^{2} + e^{x^{2}}) - 9 sin (3 x + 1)) y

= \frac{d ^{2} y}{d x ^{2}} (e^{x^{2}} + sin (3 x + 1)) + 2 \frac{d y}{d x} (2 e^{x^{2}} x + 3 cos (3 x + 1)) + (2 (2 e^{x^{2}} x^{2} + e^{x^{2}}) - 9 sin (3 x + 1)) y