展開する (x+1)^4y''(x)+2(x+1)^3y'(x)

解

展開する

(x + 1)^{4} y^{''} (x) + 2 (x + 1)^{3} y^{'} (x)

x^{4} y'' (x) + 4 x^{3} y'' (x) + 6 x^{2} y'' (x) + 4 x y'' (x) + y'' (x) + 2 x^{3} y' (x) + 6 x^{2} y' (x) + 6 x y' (x) + 2 y' (x)

解答ステップ

(x + 1)^{4} y^{''} (x) + 2 (x + 1)^{3} y^{'} (x)

(x + 1)^{4} = x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1

= (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1) y^{^{''}} (x) + 2 (x + 1)^{3} y^{^{'}} (x)

(x + 1)^{3} = x^{3} + 3 x^{2} + 3 x + 1

= (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1) y^{^{''}} (x) + 2 (x^{3} + 3 x^{2} + 3 x + 1) y^{^{'}} (x)

拡張

y^{''} (x) (x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1) : x^{4} y^{''} (x) + 4 x^{3} y^{''} (x) + 6 x^{2} y^{''} (x) + 4 x y^{''} (x) + y^{''} (x)

= x^{4} y^{''} (x) + 4 x^{3} y^{''} (x) + 6 x^{2} y^{''} (x) + 4 x y^{''} (x) + y^{''} (x) + 2 (x^{3} + 3 x^{2} + 3 x + 1) y^{'} (x)

拡張

2 y^{'} (x) (x^{3} + 3 x^{2} + 3 x + 1) : 2 x^{3} y^{'} (x) + 6 x^{2} y^{'} (x) + 6 x y^{'} (x) + 2 y^{'} (x)

= x^{4} y^{''} (x) + 4 x^{3} y^{''} (x) + 6 x^{2} y^{''} (x) + 4 x y^{''} (x) + y^{''} (x) + 2 x^{3} y^{'} (x) + 6 x^{2} y^{'} (x) + 6 x y^{'} (x) + 2 y^{'} (x)