(d^2-2d+1)y=a^xx^2

Lösung

(d^{2} - 2 d + 1) \cdot y = a^{x} \cdot x^{2}

y = c_{1} e^{x} + c_{2} x e^{x} + e^{x} \cdot \int - \frac{a ^{x} x ^{3}}{e ^{x}} d x + e^{x} x \cdot \int \frac{a ^{x} x ^{2}}{e ^{x}} d x

Schritte zur Lösung

(d^{2} - 2 d + 1) y = a^{x} x^{2}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} x e^{x} + e^{x} \cdot \int - \frac{a ^{x} x ^{3}}{e ^{x}} d x + e^{x} x \cdot \int \frac{a ^{x} x ^{2}}{e ^{x}} d x

y = c_{1} e^{x} + c_{2} x e^{x} + e^{x} \cdot \int - \frac{a ^{x} x ^{3}}{e ^{x}} d x + e^{x} x \cdot \int \frac{a ^{x} x ^{2}}{e ^{x}} d x

(d^{3} - 5 d^{2} + 8 d - 4) \cdot y = e^{2 x} + 2 e^{x} + 3 e^{- x} + 2

(d^{2} - 3 d + 4) y = sin (x) + e^{x}

(d^{2} - 2 d + 5) \cdot y = e^{2} x sin (x)

y^{^{''}} + 8 y^{^{'}} + 16 y = x^{2}

(d^{3} - 3 d + 2) \cdot y = x