solve-the-differential-equation-D-2-2D-1-y-x-2-2x-1-

Question Number 26073 by gopikrishnan005@gmail.com last updated on 19/Dec/17

s o l v e t h e d i f f e r e n t i a l e q u a t i o n (D^{2} + 2 D + 1) y = x^{2} + 2 x + 1

Commented by gopikrishnan005@gmail.com last updated on 20/Dec/17

p l s e x p l a i n

Commented by abdo imad last updated on 21/Dec/17

\Leftrightarrow \frac{d^{2} y}{{d x}^{2}} + 2 \frac{d y}{d x} + y = x^{2} + 2 x + 1 t h e s o l u t i o n o f t h i s E D i s

y_{g} = y_{h} + y_{p} . . y_{h} i s t h e h o m o g e n s^{t} a n d y_{p} p a r t i c u l a r s^{t}

E H - - > \frac{d^{2} y}{{d x}^{2}} + 2 \frac{d y}{d x} + 1 = 0 h a v e t h e c a r a c t e r i s t i c e q u a t i o n

x^{2} + 2 x + 1 = 0 \Leftrightarrow {(x + 1)}^{2} = 0 \Leftrightarrow x = - 1 \Rightarrow y_{h} = (α x + β) e^{- x} (- 1 i s d o u b l e r o o t)

f o r y_{p} w e p u t y = {a x}^{4} + {b x}^{3} + {c x}^{2} + d x + e \Rightarrow

\frac{d y}{d x} = 4 {a x}^{3} + 3 {b x}^{2} + 2 c x + d a n d \frac{d^{2} y}{{d x}^{2}} = 12 {a x}^{2} + 6 b x + 2 c

w e f i n d a f t e r c a l c u l u s a = 0 . . b = 0 \dots c = 1 . . d = - 2 \dots e = 3

a n d y_{g} = (α x + β) e^{- x} + x^{2} - 2 x + 3 .