solve-y-3y-2y-sinx-x-

Question Number 98657 by mathmax by abdo last updated on 15/Jun/20

solve y^{^{″}} - {3 y}^{^{'}} + 2 y = \frac{sinx}{x}

Answered by maths mind last updated on 15/Jun/20

y^{″} - 3 y^{'} + 2 y = 0

\Rightarrow y = {a e}^{x} + {b e}^{2 x}

l e t y = {k e}^{x} p a r t i c u l a r s o k u t i o n

\Rightarrow y^{'} = (k^{'} + k) e^{x}, y^{″} = (k^{″} + 2 k^{'} + k) e^{x} \Rightarrow

(k^{″} - k^{'}) e^{x} = \frac{s i n (x)}{x}

\Rightarrow k^{″} - k^{'} = \frac{s i n (x) e^{- x}}{x} = I m {\frac{e^{(i - 1) x}}{x}}

k^{'} - k = \int \frac{e^{(i - 1) x}}{x} d x = \int \frac{e^{u}}{u} d u = E i (u) = E i ((i - 1) x)

\Rightarrow k^{'} - k = E_{i} ((i - 1) x)

k = {s e}^{x} \Rightarrow s^{'} = \int e^{- x} E_{i} ((i - 1) x)

= [- e^{- x} E_{i} (i - 1) x)] + \int e^{- x} . \frac{e^{(i - 1) x}}{x} d x

= E_{i} ((i - 2) x)

k = e^{x} E_{i} ((i - 2) x))

Y_{p} = I m {e^{2 x} E_{i} ((i - 2) x))}

e^{2 x} I m {\int_{- \infty}^{x} \frac{e^{(i - 2) x}}{x} d x}

= e^{2 x} I m {\int \frac{e^{- 2 x} (c o s (x) + i s i n (x))}{x} d x}

= e^{2 x} [\frac{E_{i} ((i - 2) x) - E_{i} ((- i - 2) x)}{2 i}]

S = {a e}^{x} + {b e}^{2 x} + e^{2 x} [\frac{E_{i} ((i - 2) x) - E_{i} ((- i - 2) x)}{2 i}]

Commented by mathmax by abdo last updated on 15/Jun/20

thank you sir .

Answered by mathmax by abdo last updated on 15/Jun/20

(he) \to y^{^{″}} - {3 y}^{^{'}} + 2 y = 0 \to r^{2} - 3 r + 2 = 0 \Rightarrow r^{2} - 1 - 3 r + 3 = 0 \Rightarrow

(r - 1) (r + 1) - 3 (r - 1) = 0 \Rightarrow (r - 1) (r - 2) = 0 \Rightarrow r = 1 or r = 2

\Rightarrow y_{h} = a e^{x} + b e^{2 x} = {au}_{1} + b u_{2}

W (u_{1}, u_{2}) = | \begin{matrix} e^{x} e^{2 x} \\ e^{x} {2 e}^{2 x} \end{matrix} | = {2 e}^{3 x} - e^{3 x} = e^{3 x} \neq 0

W_{1} = | \begin{matrix} 0 e^{2 x} \\ \frac{sinx}{x} {2 e}^{2 x} \end{matrix} | = - e^{2 x} \frac{sinx}{x}

W_{2} = | \begin{matrix} e^{x} 0 \\ e^{x} \frac{sinx}{x} \end{matrix} | = e^{x} \frac{sinx}{x}

v_{1} = \int \frac{W_{1}}{W} dx = \int \frac{- e^{2 x} \frac{sinx}{x}}{e^{3 x}} dx = - \int e^{- 5 x} \frac{sinx}{x} dx

v_{2} = \int \frac{W_{2}}{W} dx = \int \frac{e^{x} \frac{sinx}{x}}{e^{3 x}} = \int e^{- 2 x} \frac{sinx}{x} dx \Rightarrow

y_{p} = u_{1} v_{1} + u_{2} v_{2} = - e^{x} \int^{x} e^{- 5 t} \frac{sint}{t} dt + e^{2 x} \int^{x} e^{- 2 t} \frac{sint}{t} dt

the general solution is y = y_{h} + y_{p}

= {ae}^{x} + {be}^{2 x} - e^{x} \int^{x} e^{- 5 t} \frac{sint}{t} dt + e^{2 x} \int^{x} e^{- 2 t} \frac{sint}{t} dt