erweitern (x+1)^4y''(x)+2(x+1)^3y'(x)

Lösung

erweitern

(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)

Schritte zur Lösung

(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)

Multipliziere aus

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)

Multipliziere aus

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)

e x p an d e^{x yz} - e^{x y (z - 1)}

e x p an d \frac{12 ( s + 5 )}{( s + 1 ) ^{2} ( s + 6 ) ( s ^{2} + 4 s + 5 )}

e x p an d ln (3) \cdot 3^{x^{2} + 2 y} (2 x + 2 \frac{d y}{d x})

e x p an d \frac{3}{20} \cdot (16 πh - π h^{2})

e x p an d \frac{9 x + 12}{( 2 x + 3 ) ( x - 5 )}