Characteristic equation given by
x 3 − 2 x 2 + x − sinh x − 2019 = 0 x^3 - 2 x^2 +x - \sinh x-2019= 0 x 3 − 2 x 2 + x − sinh x − 2019 = 0
Th general solution
y ′ ′ ′ − 2 y ′ ′ + y = 0 R e w r i t e t h e e q u a t i o n w i t h y = e γ t ( ( e γ t ) ) ′ ′ ′ − 2 ( ( e γ t ) ) ′ ′ + e γ t = 0 e γ t ( γ 3 − 2 γ 2 + 1 ) = 0 γ = 1 , γ = 1 + 5 2 , γ = 1 − 5 2 F o r n o n r e p e a t e d r e a l r o o t s γ 1 , γ 2 , . . . , γ n , t h e g e n e r a l s o l u t i o n t a k e s t h e f o r m : y = c 1 e γ 1 t + c 2 e γ 2 t + . . . + c n e γ n t c 1 e t + c 2 e 1 + 5 2 t + c 3 e 1 − 5 2 t y = c 1 e t + c 2 e ( 1 + 5 ) t 2 + c 3 e ( 1 − 5 ) t 2 y'''\:-2y''\:+y=0\\
\mathrm{Rewrite\:the\:equation\:with\:}y=e^{γt}\\
\left(\left(e^{γt}\right)\right)'''\:-2\left(\left(e^{γt}\right)\right)''\:+e^{γt}=0\\
e^{γt}\left(γ^3-2γ^2+1\right)=0\\
γ=1,\:γ=\frac{1+\sqrt{5}}{2},\:γ=\frac{1-\sqrt{5}}{2}\\
\mathrm{For\:non\:repeated\:real\:roots\:}γ_1,\:γ_2,\:...,\:γ_n\mathrm{,\:the\:general\:solution\:takes\:the\:form:}\\
y=c_1e^{γ_1\:t}+c_2e^{γ_2\:t}+...+c_ne^{γ_n\:t}\\
c_1e^t+c_2e^{\frac{1+\sqrt{5}}{2}t}+c_3e^{\frac{1-\sqrt{5}}{2}t}\\
y=c_1e^t+c_2e^{\frac{\left(1+\sqrt{5}\right)t}{2}}+c_3e^{\frac{\left(1-\sqrt{5}\right)t}{2}}\\ y ′′′ − 2 y ′′ + y = 0 Rewrite the equation with y = e γ t ( ( e γ t ) ) ′′′ − 2 ( ( e γ t ) ) ′′ + e γ t = 0 e γ t ( γ 3 − 2 γ 2 + 1 ) = 0 γ = 1 , γ = 2 1 + 5 , γ = 2 1 − 5 For non repeated real roots γ 1 , γ 2 , ... , γ n , the general solution takes the form : y = c 1 e γ 1 t + c 2 e γ 2 t + ... + c n e γ n t c 1 e t + c 2 e 2 1 + 5 t + c 3 e 2 1 − 5 t y = c 1 e t + c 2 e 2 ( 1 + 5 ) t + c 3 e 2 ( 1 − 5 ) t
Particular solution
y p = sinh t + 2019 D 3 − 2 D 2 + D = 1 2 [ 1 + t − 1 6 ( sinh t ) ] ∴ y = c 1 e t + c 2 e ( 1 + 5 ) t 2 + c 3 e ( 1 − 5 ) t 2 + 1 2 [ 1 + t − 1 6 ( sinh t ) ] y_p=\frac{\sinh t +2019}{D^3-2D^2+D}\\
=\frac{1}{2}[1+t-\frac{1}{6}(\sinh t)]\\
\therefore y= c_1e^t+c_2e^{\frac{\left(1+\sqrt{5}\right)t}{2}}+c_3e^{\frac{\left(1-\sqrt{5}\right)t}{2}}+\frac{1}{2}[1+t-\frac{1}{6}(\sinh t)]\\ y p = D 3 − 2 D 2 + D s i n h t + 2019 = 2 1 [ 1 + t − 6 1 ( sinh t )] ∴ y = c 1 e t + c 2 e 2 ( 1 + 5 ) t + c 3 e 2 ( 1 − 5 ) t + 2 1 [ 1 + t − 6 1 ( sinh t )]
#339914 on Dec 2023
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