D-2-4D-y-x-2-e-2x-

Question Number 103138 by bemath last updated on 13/Jul/20

(D^{2} - 4 D) y = x^{2} e^{2 x}

Commented by bemath last updated on 14/Jul/20

Commented by Israrchajk724 last updated on 13/Jul/20

please any one tell me can we convert math editor to word \dots ?

Answered by mathmax by abdo last updated on 14/Jul/20

y^{^{″}} - {4 y}^{^{'}} = x^{2} e^{2 x}

(h) \to r^{2} - 4 r = 0 \Rightarrow r (r - 4) = 0 \Rightarrow r = o or r = 4 \Rightarrow y_{h} = a + {be}^{4 x}

= {au}_{1} + {bu}_{2}

W (u_{1}, u_{2}) = | \begin{matrix} 1 e^{4 x} \\ 0 {4 e}^{4 x} \end{matrix} | = {4 e}^{4 x} \neq 0

W_{1} = | \begin{matrix} o e^{4 x} \\ x^{2} e^{2 x} {4 e}^{4 x} \end{matrix} | = - x^{2} e^{6 x}

W_{2} = | \begin{matrix} 1 o \\ 1 x^{2} e^{2 x} \end{matrix} | = x^{2} e^{2 x}

v_{1} = \int \frac{w_{1}}{W} dx = \int \frac{- x^{2} e^{6 x}}{{4 e}^{4 x}} dx = - \frac{1}{4} \int x^{2} e^{2 x} dx

\int x^{2} e^{2 x} dx = \frac{x^{2}}{2} e^{2 x} - \int 2 x \times \frac{1}{2} e^{2 x} dx = \frac{x^{2}}{2} e^{2 x} - \int {xe}^{2 x} dx

= \frac{x^{2}}{2} e^{2 x} - {\frac{x}{2} e^{2 x} - \int \frac{1}{2} e^{2 x} dx} = \frac{x^{2}}{2} e^{2 x} - \frac{x}{2} e^{2 x} + \frac{1}{4} e^{2 x} \Rightarrow

v_{1} = - \frac{1}{4} {\frac{x^{2}}{2} e^{2 x} - \frac{x}{2} e^{2 x} + \frac{1}{4} e^{2 x}}

v_{2} = \int \frac{w_{2}}{w} dx = \int \frac{x^{2} e^{2 x}}{{4 e}^{4 x}} dx = \frac{1}{4} \int x^{2} e^{- 2 x} dx

{4 v}_{2} = - \frac{x^{2}}{2} e^{- 2 x} - \int 2 x \times (- \frac{1}{2}) e^{- 2 x} dx = - \frac{x^{2}}{2} e^{- 2 x} + \int {xe}^{- 2 x} dx

= - \frac{x^{2}}{2} e^{- 2 x} - \frac{x}{2} e^{- 2 x} + \int \frac{1}{2} e^{- 2 x} dx = - \frac{x^{2}}{2} e^{- 2 x} - \frac{x}{2} e^{- 2 x} - \frac{1}{4} e^{- 2 x} \Rightarrow

v_{2} = \frac{1}{4} {- \frac{x^{2}}{2} - \frac{x}{2} - \frac{1}{4}} e^{- 2 x}

y_{p} = u_{1} v_{1} + u_{2} v_{2} = - \frac{1}{4} {\frac{x^{2}}{2} - \frac{x}{2} + \frac{1}{4}} e^{2 x} + e^{4 x} \times \frac{1}{4} {- \frac{x^{2}}{2} - \frac{x}{2} - \frac{1}{4}} e^{- 2 x}

= \frac{1}{4} {- \frac{x^{2}}{2} + \frac{x}{2} - \frac{1}{4}} e^{2 x} + \frac{1}{4} {- \frac{x^{2}}{2} - \frac{x}{2} - \frac{1}{4}} e^{2 x}

= \frac{1}{4} {- x^{2} - \frac{1}{2}} e^{2 x} = - \frac{1}{4} (x^{2} + \frac{1}{2}) e^{2 x}

y = y_{h} + y_{p}