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Question Number 202651 by hardmath last updated on 31/Dec/23

Commented by Frix last updated on 31/Dec/23

$y = \frac{3 \cos x + \sin x}{10} + c_{1} e^{2 x} + c_{2} e^{x}$
	Answered by Mathspace last updated on 31/Dec/23

	$h \to r^{2} - 3 r + 2 = 0$ $Δ = 9 - 8 = 1 \Rightarrow r_{1} = \frac{3 + 1}{2} = 2$ $r_{2} = \frac{3 - 1}{2} = 1 \Rightarrow y_{h} = {a e}^{x} + {b e}^{2 x}$ $= {a u}_{1} + {b u}_{2}$ $w (u_{1}, u_{2}) = \| \begin{matrix} u_{1} u_{2} \\ u_{1}^{^{'}} u_{2}^{^{'}} \end{matrix} \|$ $= \| \begin{matrix} e^{x} e^{2 x} \\ e^{x} 2 e^{2 x} \end{matrix} \| = 2 e^{3 x} - e^{3 x} = e^{3 x}$ $w_{1} = \| \begin{matrix} 0 e^{2 x} \\ s i n x 2 e^{2 x} \end{matrix} \| = - e^{2 x} s i n x$ $w_{2} = \| \begin{matrix} e^{x} 0 \\ e^{x} s i n x \end{matrix} \| = e^{x} s i n x$ $v_{1} = \int \frac{w_{1}}{w} d x = - \int \frac{e^{2 x} s i n x}{e^{3 x}} d x$ $= - \int e^{- x} s i n x d x$ $= - I m (\int e^{- x + i x} d x)$ $w e h a v e \int e^{(- 1 + i) x} d x$ $= \frac{1}{- 1 + i} e^{(- 1 + i) x}$ $= - \frac{1}{1 - i} e^{(- 1 + i) x} = - \frac{1 + i}{2} e^{- x} (c o s x + i s i n x)$ $= - \frac{e^{- x}}{2} (c o s x + i s i n x - i c o s x - s i n x)$ $\Rightarrow v_{1} = \frac{e^{- x}}{2} (s i n x - c o s x)$ $v_{2} = \int \frac{w_{2}}{w} d x = \int \frac{e^{x} s i n x}{e^{3 x}} d x$ $\int e^{- 2 x} s i n x d x = I m (\int e^{(- 2 + i) x} d x)$ $\int e^{(- 2 + i) x} d x = \frac{1}{- 2 + i} e^{(- 2 + i) x}$ $= - \frac{1}{2 - i} e^{(- 2 + i) x} = - \frac{2 + i}{5} e^{- 2 x} (c o s x + i s i n x)$ $= - \frac{e^{- 2 x}}{5} (2 c o s x + 2 i s i n x + i c o s x - s i n x)$ $\Rightarrow v_{2} = - \frac{e^{- 2 x}}{5} (2 s i n x + c o s x)$ $p a r t i c u l a r s o l u t i o n i s$ $y_{p} = u_{1} v_{1} + u_{2} v_{2}$ $= e^{x} . \frac{e^{- x}}{2} (s i n x - c o s x)$ $- e^{2 x} . \frac{e^{- 2 x}}{5} (2 s i n x + c o s x)$ $= \frac{1}{2} s i n x - \frac{1}{2} c o s x - \frac{2}{5} s i n x$ $- \frac{1}{5} c o s x = \frac{1}{10} s i n x - \frac{7}{10} c o s x$ $s o y_{g} = y_{p} + y_{h}$ $= \frac{1}{10} s i n x - \frac{7}{10} c o s x + {a e}^{x} + {b e}^{2 x}$
	Commented by Frix last updated on 03/Jan/24

	$The method is ok but the result is wrong .$
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