### Author Topic: FE-P3  (Read 11983 times)

#### Victor Ivrii

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##### FE-P3
« on: December 14, 2018, 07:54:21 AM »
Typed solutions only. No uploads

Find the general solution of
\begin{equation*}
y'''-2y'' -y '+2y = \frac{12e^{2t}}{e^t+1}.
\end{equation*}
Hint: All roots are integers (or complex integers).

#### Qi Cui

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##### Re: FE-P3
« Reply #1 on: December 14, 2018, 09:23:35 AM »
$$Homo: r^3 -2r^2-r+2=0$$
$$(r+1)(r-1)(r-2)=0$$
$$r_1=1, r_2=-1, r_3=2$$
$$\therefore y_c(t)=c_1e^t+c_2e^{-t}+c_3e^{2t}$$
$$W = \left| \begin {array}{ccc} {e^t}&e^{-t}&e^{2t}\\ e^t& -e^{-t}&2e^{2t}\\ e^t&e^{-t}& 4e^{2t} \end {array} \right| = -6e^{2t}$$
$$W_1 = \left| \begin {array}{ccc} {0}&e^{-t}&e^{2t}\\ 0& -e^{-t}&2e^{2t}\\ 1&e^{-t}& 4e^{2t} \end {array} \right| = 3e^{t}$$
$$W_2 = \left| \begin {array}{ccc} {e^t}&0&e^{2t}\\ e^t&0&2e^{2t}\\ e^t&1& 4e^{2t} \end {array} \right| = e^{3t}$$
$$W_3 = \left| \begin {array}{ccc} {e^t}&e^{-t}&0\\ e^t& -e^{-t}&0\\ e^t&e^{-t}& 1 \end {array} \right| = -2$$
substitute above into the formula:
$$y_p(t)=y_1{\int}{{w_1(s)}{g(s)}\over{W(s)}} ds+y_2{\int}{{w_2(s)}{g(s)}\over{W(s)}}ds+y_3{\int}{{w_2(s)}{g(s)}\over{W(s)}}ds$$
$$=-e^t{\int}{{6e^s}\over{e^s+1}} ds-e^{-t} {\int}{{2e^{3s}}\over{e^s+1}}ds+e^{2t}{\int}{{4}\over{e^s+1}}ds$$
$$=-6e^tln|e^t+1|-e^{-t}ln|e^t+1|-e^{-t}(e^t-1)^2-e^{2t}4ln|e^{-x}+1|$$
$$y(t)=y_c(t)+y_p(t)$$
« Last Edit: December 14, 2018, 10:08:45 AM by Qi Cui »

#### Doris Zhuomin Jia

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##### Re: FE-P3
« Reply #2 on: December 14, 2018, 09:50:28 AM »
homo: r3 - 2r2 - r+2 = 0
(r-1)(r+1)(r-2) =0   r = 1, -1, 2
y1= et   y2=e-t   y3=e2t

W(t,y) = et       e-t     e2t
et      -e-t    2e2t
et       e-t    4e2t
= et(-e-t·4e2t-e-t·2e2t) - e-t(et· 4e2t- et· 2e2t）+ e2t(et·e-t+et·e-t)
= -6e2t - 2e2t + 2e2t =-6e2t
W1 = e-t·2e2t+e-t·e2t= et
W2 = et· 2e2t - et·e2t  = e3t
W3 = et· -e-t -  et·e-t = -2

u1 = ∫ [(2e2t/et +1)·et]/-6e2t = -2ln|et+1| +C1
u2 = ∫[(2e2t/et +1)·e3t]/-6e2t = 2et-e2t+3 -2ln|et+1| +C2
u3 = ∫[(2e2t/et +1)·(-2)]/-6e2t =4(-ln|et+1|+t)+C3

y= u1y1+u2y2+u3y3
y= et(-2ln|et+1| +C1)+e-t(2et-e2t+3 -2ln|et+1| +C2)+e2t(4(-ln|et+1|+t)+C3)
« Last Edit: December 14, 2018, 10:10:56 AM by Doris Zhuomin Jia »

#### Jiexuan Wei

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##### Re: FE-P3
« Reply #3 on: December 14, 2018, 12:16:57 PM »
forget to see 'readme before posting', sorry
« Last Edit: December 14, 2018, 12:20:11 PM by Jiexuan Wei »

#### Chengyin Ye

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##### Re: FE-P3
« Reply #4 on: December 14, 2018, 12:52:25 PM »
I think that instead of writing e-t(2et-e2t+3 -2ln|et+1| +C2), we should write e-t(2et -2ln|et+1| +C2) because e-t(-e2t) and 3et  are in the homogeneous solution.

#### Qinger Zhang

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##### Re: FE-P3
« Reply #5 on: December 17, 2018, 08:12:41 PM »

$$W_2 = \left| \begin {array}{ccc} {e^t}&0&e^{2t}\\ e^t&0&2e^{2t}\\ e^t&1& 4e^{2t} \end {array} \right| = e^{3t}$$
I think W2 is negative.
$$W_2 = \left| \begin {array}{ccc} {e^t}&0&e^{2t}\\ e^t&0&2e^{2t}\\ e^t&1& 4e^{2t} \end {array} \right| = -e^{3t}$$
$$y_p(t)=y_1{\int}{{w_1(s)}{g(s)}\over{W(s)}} ds+y_2{\int}{{w_2(s)}{g(s)}\over{W(s)}}ds+y_3{\int}{{w_2(s)}{g(s)}\over{W(s)}}ds$$
$$=-e^t{\int}{{6e^s}\over{e^s+1}} ds+e^{-t} {\int}{{2e^{3s}}\over{e^s+1}}ds+e^{2t}{\int}{{4}\over{e^s+1}}ds$$
$$=-6e^tln|e^t+1|+e^{-t}ln|e^t+1|-e^{-t}(e^t-1)^2-e^{2t}4ln|e^{-x}+1|$$
$$y(t)=c_1e^t+c_2e^{-t}+c_3e^{2t}-6e^tln|e^t+1|+e^{-t}ln|e^t+1|-e^{-t}(e^t-1)^2-e^{2t}4ln|e^{-x}+1|$$
« Last Edit: December 17, 2018, 08:19:25 PM by Qinger Zhang »

#### Victor Ivrii

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##### FE-P3 official
« Reply #6 on: December 20, 2018, 01:59:33 AM »
Writing characteristic equation: $L(k):= k^3-2k^2-k+2=0$, with $L(k)=(k-2)k^2 -(k-2)=(k-2)(k^2-1)$; then $k_1=1, k_2=-1, k_3=2$. Then

y^*= C_1e^{t} + C_2 e^{-t}  + C_3 e^{2t}
\label{eq-3-1}

is a general solution to the homogeneous equation.

We are looking for solution to the inhomogeneous equation as (\ref{eq-3-1}) with unknown functions $C_1,C_2,C_3$ s.t.
\begin{align*}
&\left\{\begin{aligned}
&C_1' e^{t} + C_2' e^{-t}+ C_3' e^{2t}=0,\\
&C_1' e^{t} - C_2' e^{-t} + 2C_3' e^{2t}=0,\\
&C_1' e^{t} + C_2' e^{-t}+4 C_3' e^{2t}=\frac{12e^{2t}}{e^t+1};
\end{aligned}\right.\implies\\[4pt]
&\left\{\begin{aligned}
2&C_1' e^{t} + 3C_3' e^{2t}=0,\\
&C_1' e^{t} - C_2' e^{-t} + 2C_3' e^{2t}=0,\\
2&C_1' e^{t} +  6 C_3' e^{2t}=\frac{12e^{2t}}{e^t+1};
\end{aligned}\right.\implies\\[4pt]
&C_3'e^{2t}=\frac{4e^{2t}}{e^t+1}, \qquad C_1'e^{t}= -\frac{6e^{2t}}{e^t+1},\qquad C_2'e^{-t}=\frac{2e^{2t}}{e^t+1}\implies\\[4pt]
&C_1=-\int \frac{6e^{t}\,dt}{e^t+1}= -6 \ln (e^t+1) +c_1,\\[4pt]
&C_2= \int \frac{2e^{3t}\,dt}{e^t+1}= \int \Bigl[2e^{2t}- 2 e^{t}\Bigr] \,dt +\int  \frac{2e^{t}\,dt}{e^t+1}
=e^{2t} -2e^{t} +2\ln (e^t+1)+c_2,\\[4pt]
&C_3=\int \frac{4\,dt}{e^t+1}=\int \frac{4e^{-t}\,dt}{1+e^{-t}}= -4\ln (1+e^{-t})+c_3= -4\ln (e^t+1)+4t +c_3.
\end{align*}
Then
\begin{align*}
y= &\bigl[-6 \ln (e^t+1) +c_1\bigr]e^{t}+\bigl[e^{2t} -2e^{t} +2\ln (e^t+1)+c_2\bigr]e^{-t}+
\bigl[-4\ln (e^t+1)+4t +c_3\bigr]e^{2t}=\\
&-6  e^{t}\ln (e^t+1)+ 2e^{-t}\ln (e^t+1)
-4e^{2t}\ln (e^t+1)+4te^{2t}-2 +
c_1e^{t} + c_2 e^{-t}  + c_3 e^{2t}
\end{align*}
with $c_1:= c_1+1$ in the last transition.