y′+y+7=0


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Question Number 111644 by weltr last updated on 04/Sep/20

$y^{'} + y + 7 = 0$
Commented by mohammad17 last updated on 04/Sep/20

$(D + 1) y = - 7$ $r + 1 = 0 \Rightarrow r = - 1$ $y_{h} = C e^{- x}$ $l e t : y_{p} = a \Rightarrow y_{p}^{^{'}} = 0$ $(0 + a) = - 7 \Rightarrow a = - 7$ $y = y_{h} + y_{p}$ $y = C e^{- x} - 7$ $b y :≪ m o h a m m a d ≫$
Commented by weltr last updated on 04/Sep/20

$t h a n k y o u$
Commented by mohammad17 last updated on 04/Sep/20
$method 2 (D+1)y=−7 (r+1)=0⇒r=−1 y_h =C e^(−x) let: y_p = ((1/(1+D)))(−7) y_p = {1−D+....}(−7) y_p =−7+0+...⇒y_p =−7 y=y_h +y_p y=C e^(−x) −7 ≪mohammad≫$
$m e t h o d 2$ $(D + 1) y = - 7$ $(r + 1) = 0 \Rightarrow r = - 1$ $y_{h} = C e^{- x}$ $l e t : y_{p} = (\frac{1}{1 + D}) (- 7)$ $y_{p} = {1 - D + . . . .} (- 7)$ $y_{p} = - 7 + 0 + . . . \Rightarrow y_{p} = - 7$ $y = y_{h} + y_{p}$ $y = C e^{- x} - 7$ $≪ m o h a m m a d ≫$
Commented by mohammad17 last updated on 04/Sep/20

$y o u a r e w e l c o m e$
	Answered by mathmax by abdo last updated on 04/Sep/20

	$\to r + 1 = 0 \Rightarrow r = - 1 \Rightarrow y (x) = {ae}^{- x} + b$ $y^{^{'}} = - {ae}^{- x} e \Rightarrow - {ae}^{- x} + {ae}^{- x} + b + 7 = 0 \Rightarrow b = - 7 \Rightarrow$ $y (x) = {ae}^{- x} - 7$
	Answered by Aziztisffola last updated on 04/Sep/20

	$let y = u v \Rightarrow y^{'} = u^{'} v + {u v}^{'}$ $u^{'} v + {u v}^{'} + u v = - 7 \Leftrightarrow {u v}^{'} + v (u^{'} + u) = - 7$ $u^{'} + u = 0 \Rightarrow \frac{u^{'}}{u} = - 1 \Rightarrow \int \frac{u^{'}}{u} dx = \int - dx$ $\ln (u) = - x + C \Rightarrow u = e^{- x + C} = k e^{- x}$ $k e^{- x} v^{'} = - 7 \Rightarrow v^{'} = - \frac{7}{k} e^{x} \Rightarrow v = - \frac{7}{k} e^{x} + C$ $y = k e^{- x} (- \frac{7}{k} e^{x} + C) = {K e}^{- x} - 7 (K \in R)$
	Answered by john santu last updated on 04/Sep/20
	$HE: λ+1=0; λ=−1 ⇒y_(h ) = C.e^(−x) particular solution y_p = Ax+B comparing coeff y′=A, ⇒A+Ax+B+7 = 0 Ax + (B+A+7) = 0 { ((A=0)),((B=−7)) :} ⇒y_p = −7 general solution y_g = C.e^(−x) −7.$
	$H E : λ + 1 = 0; λ = - 1 \Rightarrow y_{h} = C . e^{- x}$ $p a r t i c u l a r s o l u t i o n$ $y_{p} = A x + B$ $c o m p a r i n g c o e f f$ $y^{'} = A, \Rightarrow A + A x + B + 7 = 0$ $A x + (B + A + 7) = 0$ ${\begin{cases} A = 0 \\ B = - 7 \end{cases} \Rightarrow y_{p} = - 7$ $g e n e r a l s o l u t i o n$ $y_{g} = C . e^{- x} - 7 .$
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