Problem 2 (main sitting)

[Victor Ivrii]

Consider equation
\begin{equation}
y'''+4y''+y'-6y=24e^{t}.
\label{2-1}
\end{equation}
(a) Write a differential equation for Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.

(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).

(c) Find the general solution of (\ref{2-1}).

[Yiheng Bian]

(a):
$$
W=ce^{-\int{p(t)}}dt\\
W=ce^{-\int{4}dt}\\
W=ce^{-4t}
$$



(b):
We can get:
$$
r^3+4r^2+r-6=0\\
(r-1)(r+2)(r+3)=0\\
r_1=1,r_2=-2,r_3=-3
$$
Therefore:
$$
y=c_1e^t+c_2e^{-2t}+c_3e^{-3t}
$$
$$
\begin{vmatrix}
e^t & e^{-2t} & e^{-3t} \\
e^t & -2e^{-2t} & -3e^{-3t} \\
e^t & 4e^{-2t} & 9e^{-3t}
\end{vmatrix}=-12e^{-4t}
$$
compare with (a) we get
$$
c=-12
$$


(c):
Let:
$$
y_p(t)=Ate^t\\
y'=A(e^t+te^t)\\
y''=A(2e^t+te^t)\\
y'''=A(3e^t+te^t)
$$
We take these into equation and get:
$$
A(3e^t+te^t)+4A(2e^t+te^t)+A(e^t+te^t)-6Ate^t=24e^t\\
12Ae^t=24e^t\\
A=2
$$
So:
$$
y_p(t)=2te^t
$$
Therefore:
$$
y=c_1e^t+c_2e^{-2t}+c_3e^{-3t}+2te^t
$$

OK. V.I.

[xuanzhong]

a)Write equation for Wronskian of y1,y2,y3.
$$
W=ce^{-\int p(t)dt}=ce^{-\int 4dt}=ce^{-4t}
$$
b)Find fundamental system of solutions for honogenuous equation, and find Wronskian. Comapare with (a).
$$
r^3+4r^2+r-6=0
$$
$$
(r-1)(r+2)(r+3)=0
$$
$$
r=1, r=-2, r=-3
$$
$$
y_c(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}
$$
$$
W=\begin{vmatrix}
e^t & e^{-2t} & e^{-3t}\\
e^t & -2e^{-2t} & -3e^{-3t}\\
e^t & 4e^{-2t} & 9e^{-3t}\\
\end{vmatrix}
= 12e^{-4t}
$$
similar solution to (a), but $c=12$

(c)Find the general solution.
$$
y_p(t)=Ate^t
$$
$$
y^\prime=Ae^t+Ate^t=Ae^t(1+t)
$$
$$
y^{\prime\prime}=2Ae^t+Ate^t=Ae^t(2+t)
$$
$$
y^{\prime\prime\prime}=3Ae^t+Ate^t=Ae^t(3+t)
$$
$$
y^{\prime\prime\prime}+4y^{\prime\prime}+y^\prime-6y=24e^t
$$
$$
Ae^t(3+t+8+4t+1+t-6t)=24e^t
$$
$$
12Ae^t=24e^t
$$
$$
A=2
$$
Hence, $y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}+2te^t$

[Yuying Chen]

$\text(a)\\$
$W=ce^{-\int p(t)dt}=ce^{-\int 4dt}=ce^{-4t}\\$
$\text(b)\\$
$r^3+4r^2+r-6=0\\$
$(r-1)(r^2+5r+6)=0\\$
$r=1, r=-2, r=-3\\$
$\therefore y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}\\$
$W=\begin{vmatrix}
e^t & e^{-2t} & e^{-3t}\\
e^t &  -2e^{-2t} & -3e^{-3t}\\
e^t & 4e^{-2t} & 9e^{-3t}\\
\end{vmatrix}=-12e^{-4t}\\$
$\text{$\therefore c=-12$ compare with (a)}\\$
$\text(c)\\$
$y_p(t)=Ate^t\\$
$y_p^{\prime}(t)=Ae^t+Ate^t\\$
$y_p^{\prime\prime}(t)=2Ae^t+Ate^t\\$
$y_p^{\prime\prime\prime}(t)=3Ae^t+Ate^t\\$
$3Ae^t+Ate^t+8Ae^t+4Ate^t+Ae^t+Ate^t-6Ate^t=12Ae^t\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad 12A=24\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad A=2$
$\therefore y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}+2te^t$

[Xinyu Jing]

(𝑎)
𝑊=$𝑐𝑒^{−∫𝑝(𝑡)𝑑𝑡}$=$𝑐𝑒^{−∫4𝑑𝑡}$=$𝑐𝑒^{−4𝑡}$
(𝑏)
$𝑟^{3}+4𝑟^{2}+𝑟−6$=0
(𝑟−1)($𝑟^{2}$+5𝑟+6)=0
𝑟=1,𝑟=−2,𝑟=−3
∴𝑦(𝑡)=$𝑐_{1}𝑒^{𝑡}+𝑐_{2}𝑒^{−2𝑡}+𝑐_{3}𝑒^{−3𝑡}$
 
W=\begin{vmatrix}
       e^{t} & e^{-2t} &  e^{-3t} \\
       e^{t} & -2e^{-2t} & -3e^{-3t} \\   
       e^{t} & 4e^{-2t} & 9e^{-3t} \\
     \end{vmatrix}=−12𝑒−4𝑡
∴𝑐=−12 compare with (a)
(𝑐)
𝑦𝑝(𝑡)=$𝐴𝑡𝑒^{𝑡}$
𝑦′𝑝(𝑡)=$𝐴𝑒^{𝑡}+𝐴𝑡𝑒^{𝑡}$
𝑦′′𝑝(𝑡)=$2𝐴𝑒^{𝑡}+𝐴𝑡𝑒^{𝑡}$
𝑦′′′𝑝(𝑡)=$3𝐴𝑒^{𝑡}+𝐴𝑡𝑒^{𝑡}$
$3𝐴𝑒^{𝑡}+𝐴𝑡𝑒^{𝑡}+8𝐴𝑒^{𝑡}+4𝐴𝑡𝑒^{𝑡}+𝐴𝑒^{𝑡}+𝐴𝑡𝑒^{𝑡}−6𝐴𝑡𝑒^{𝑡}=12𝐴𝑒^{𝑡}$
12𝐴=24
𝐴=2
∴𝑦(𝑡)=$𝑐_{1}𝑒^{𝑡}+𝑐_{2}𝑒^{−2𝑡}+𝑐_{3}𝑒^{−3𝑡}+2𝑡𝑒^{𝑡}$

[Ruodan Chen]

2) $y'''+4y''+y'-6y=24e^{t}$

a)

Since the coefficient of y'' is 4,

Then, Wronskain is

$w=ce^{-\int4dt}=ce^{-4t}$

b)

Use homogeneous equation to find fundamental solutions,

$y'''+4y''+y'-6y=24e^{t}$

Then $r^{3}+4r^{2}+r-6=0$

Then $r=1, r=-2, r=-3$

Then the solution is $y_{c}(t)=c_{1}e^{t}+c_{2}e^{-2t}+c_{3}e^{-3t}$

$w = \begin{array}{ccc}
e^{t} & e^{-2t} & e^{-3t}\\
e^{t} & -2e^{-2t} & -3e^{-3t}\\
e^{t} & 4e^{-2t} & 9e^{-3t}
\end{array}=-12e^{-4t}$

$w = -12e^{-4t} =ce^{-4t}$

This is consistent with what we get in part (a) with $c=-12$

c)

Use under determined coefficients method to find the general equation for y(t)

$y'''+4y''+y'-6y=24e^{t}$

$y_{p}(t)=Ate^{t}$

$y'_{p}(t)=Ae^{t}+Ate^{t}$

$y''_{p}(t)=2Ae^{t}+Ate^{t}$

$y'''_{p}(t)=3Ae^{t}+Ate^{t}$

Plug into the equation,

$3Ae^{t}+Ate^{t}+4(2Ae^{t}+Ate^{t})+Ae^{t}+Ate^{t}-6Ate^{t}=12Ae^{t}=24e^{t}$

We get A=2

$y_{p}(t)=2te^{t}$

Thus, $y(t)=c_{1}e^{t}+c_{2}e^{-2t}+c_{3}e^{-3t}+2te^{t}$

[nayan]

$\text(a)\\$

$\text{Using abels theorem,}
\\
W=ce^{-\int p(t)dt}=ce^{-\int 4dt}=ce^{-4t}\\$
$\text(b)\\$
$\text{We have the following characteristic polynomial,}\\$
$r^3+4r^2+r-6=0\\$
$(r-1)(r^2+5r+6)=0\\$
$r=1, r=-2, r=-3\\$
$\therefore y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}\\$
$W=\begin{vmatrix}
e^t & e^{-2t} & e^{-3t}\\
e^t &  -2e^{-2t} & -3e^{-3t}\\
e^t & 4e^{-2t} & 9e^{-3t}\\
\end{vmatrix}=-12e^{-4t}\\$
$\text{$\therefore c=-12$ }\\$
$\text(c)\\$
$y_p(t)=Ate^t\\$
$y_p^{\prime}(t)=Ae^t+Ate^t\\$
$y_p^{\prime\prime}(t)=2Ae^t+Ate^t\\$
$y_p^{\prime\prime\prime}(t)=3Ae^t+Ate^t\\$
$3Ae^t+Ate^t+8Ae^t+4Ate^t+Ae^t+Ate^t-6Ate^t=12Ae^t\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad 12A=24\\$
$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad A=2$
$\text{We finally have the general solution,}\\ y(t)=c_1e^t+c_2e^{-2t}+c_3e^{-3t}+2te^t$

[Mingdi Xie]

This is my solution