Problem 3 (main)

[Victor Ivrii]

(a) Find the general solution for equation
Find the general solution for equation
\begin{equation*}
y'' -2y'-3y= 16\cosh (x).
\end{equation*}
 
(b) Find solution, satisfying $y(0)=0$, $y'(0)=0$.

[Klaire]

a,
First solve the homogenous part:
\begin{aligned}
r^{2}-2r-3 &=0\\
r^{2}-2r+1&=4\\
(r-1)^{2} &=4\\
r_1 &=3 \\r_2 &=-1
\end{aligned}
So the solution to homogenous part is:
\begin{aligned}
y_c(x) =c_1e^{3x}+c_2e^{-x}
\end{aligned}
Next we solve Never use * for multiplication
\begin{aligned}
y^{\prime\prime}-2y^{\prime}-3y &=16cosh(x)\\
                                                     &=16*\frac{e^{x}+e^{-x}}{2} \\ 
                                                     &=8e^{x}+8e^{-x}
\end{aligned}
Let
\begin{aligned}
y_p(x)&=Ae^{x}+Be^{-x}*x\\
y_p^{\prime}(x) &= Ae^{x}+Be^{-x}-Bxe^{-x}\\
y_p^{\prime\prime}(x) &= Ae^{x}-Be^{-x}-Be^{-x}+Bxe^{-x}\\
                                   &= Ae^{x}-2Be^{-x}+Bxe^{-x}\\
\end{aligned}
Thus we can have
\begin{aligned}
Ae^{x}-2Be^{-x}+Bxe^{-x}-2(Ae^{x}+Be^{-x}-Bxe^{-x})-3(Ae^{x}+Be^{-x}*x) &=8e^{x}+8e^{-x}\\
-4Ae^{x}-4Be^{x} &=8e^{x}+8e^{-x}\\
A &=-2\\
B &=-2
\end{aligned}
Therefore, we can have:
\begin{aligned}
y_p(x) &=-2e^{x}-2xe^{-x}
\end{aligned}
From the above, we get
\begin{aligned}
y&=y_c(x)+y_p(x)\\
&= c_1e^{3x}+c_2e^{-x}-2e^{x}-2xe^{-x}
\end{aligned}


b,
\begin{aligned}
y &= c_1e^{3x}+c_2e^{-x}-2e^{x}-2xe^{-x}\\
y_p^{\prime} &= 3c_1e^{3x}-c_2e^{-x}-2e^x+2xe^{-x}-2e^{-x}\\
\end{aligned}
since
\begin{aligned}
y(0) &=0\\
y^{\prime} &= 0
\end{aligned}
we can have
\begin{aligned}
c_1 &=3/2\\
c_2 &=1/2
\end{aligned}
Thus, the solution is
\begin{aligned}
y &= 3/2e^{3x}+1/2e^{-x}-2e^{x}-2xe^{-x}
\end{aligned}


OK. V.I.

[Jingjing Cui]

a)
$$
y''-2y'-3y=16cosh(x)=8e^{x}+8e^{-x}\\
r^2-2r-3=0\\
(r+1)(r-3)=0\\
r_1=-1\\
r_2=3\\
y_c(x)=c_1e^{-x}+c_2e^{3x}\\
\\
y''-2y'-3y=8e^{x}\\
y_{p1}(x)=ce^x\\
y_{p1}'(x)=y_{p1}''=ce^x\\
ce^x-2ce^x-3ce^x=8e^{x}\\
-4c=8\\
c=-2\\
y_{p1}(x)=-2e^x\\
y''-2y'-3y=8e^{-x}\\
y_{p2}(x)=cxe^{-x}\\
y_{p2}'(x)=ce^{-x}-cxe^{-x}\\
y_{p2}''(x)=-ce^x-ce^{-x}+cxe^{-x}\\
-2ce^{-x}+cxe^{-x}-2ce^{-x}+2cxe^{-x}-3cxe^{-x}=8e^{-x}\\
e^{-x}(-2c-2c)+xe^{-x}(c+2c-3c)=8e^{-x}\\
-4c=8\\
c=-2\\
y_{p2}(x)=-2xe^{-x}\\
y(x)=c_1e^{-x}+c_2e^{3x}-2e^x-2xe^{-x}\\
$$

[Anyue Huang]

Find the general solution of the equation
$$
y^{\prime \prime}-2 y^{\prime}-3 y=16 \cosh x
$$
$$
\begin{aligned} r^{2}-2 r-3 &=0 \\(r-3)(r+1) &=0 \\ r=3,  r&=-1 \end{aligned}
$$
$$
\begin{array}{l}{y_{c}=c_{1} e^{3 x}+c_{2} e^{-x}} \\ {y^{\prime \prime}-2 y^{\prime}-3 y=16 \cosh x=8 e^{x}+8 e^{-x}} \\ {y^{\prime \prime}-2 y^{\prime}-3 y=8 e^{x}}\end{array}
$$

Let $y_{p}=A e^{x}\quad y^{\prime}=A e^{x} \quad y^{\prime \prime}=A e^{x}$
$$\begin{aligned} A e^{x}-2 A e^{x}-3 A e^{x} &=8 e^{x} \\-4 A e^{x} &=8 e^{x} \\ A &=-2 \end{aligned}$$
$$
\therefore y_{p}=-2 e^{x}
$$

$$
y^{\prime \prime}-2 y^{\prime}-3 y=8 e^{-x}
$$

$\operatorname{let} y_{p}(t)=A x e^{-x} \quad y^{\prime}=A e^{-x}-A x e^{-x}$
$$
\begin{aligned} y^{\prime \prime}=&-A e^{-x}-\left(A e^{-x}-A x e^{-x}\right) \\ &=-2 A e^{-x}+A x e^{-x} \end{aligned}
$$
$$
-2 A e^{-x}+A x e^{-x}-2 A e^{-x}+2 A x e^{-x}-3 A x e^{-x}=8 e^{-x}
$$
$$
\begin{aligned}-2 A-2 A &=8 \\ A &=-2 \\ y_{p} &=-2 \times e^{-x} \end{aligned}
$$
$$
y=c_{1} e^{3 x}+c_{2} e^{-x}-2 e^{x}-2 x e^{-x}
$$

[Yijin Qiang]

${y}''-{2y}'-3y=16cosh(x)
r^2-2r-3=16cosh(x)\\
r^2-2r-3=16(\frac{e^{x}+e^{-x}}{2})=8e^{x}+8e^{-x}\\
(a). Homogeneous part:\\
let\,r^2-2r-3=0\\
=(r+1)(r-3),r_{1}=-1,r_{2}=3\\
y_{c}=C_{1}e^{-x}+C_{2}e^{3x}\\
Next\,we\,solve \, {y}''-{2y}'-3y=8e^{x}\\
let y_{1}=Ae^{x}\\
then\,{y}'=Ae^{x},{y}''=Ae^{x}\\
{y}''-{2y}'-3y=Ae^{x}-2(Ae^{x})-3(Ae^{x})\\
(A-2A-3A)e^{x}=8e^{x}\\
A=-2,y_{1}=-2e^{x}\\
now\,let\,{y}''-{2y}'-3y=8e^{-x}\\
let\, y_{p2}=Bxe^{-x}\,Since Be^{-x}=8e^{-x}\\
then\,{y}'=Be^{-x}-Bxe^{-x},{y}''=-Be^{-x}-Be^{-x}+Bxe^{-x}\\
{y}''-{2y}'-3y=-Be^{-x}-Be^{-x}+Bxe^{-x}-2(Be^{-x}-Bxe^{-x})-3Bxe^{-x}=8e^{-x}\\
B=-2,y_{p2}=-2xe^{-x}\\
y(x)=y_{c}+y_{p1}+y_{p2}=C_{1}e^{-x}+C_{2}e^{3x}-2e^{x}-2xe^{-x}\\
(b)y(x)=C_{1}e^{-x}+C_{2}e^{3x}-2e^{x}-2xe^{-x}\\
y(x)'=-C_{1}e^{-x}+3C_{2}e^{3x}-2e^{x}-2e^{-x}+2xe^{-x}\\
let\,y(0)'=0\\
-C_{1}+3C_{2}-2-2+0=0,-C_{1}+3C_{2}=4\\
let\,y(0)=0\\
C_{1}+C_{2}-2=0,C_{1}+C_{2}=2\\
C_{2}=\frac{2}{3},C_{1}=\frac{1}{2}\\
y=\frac{1}{2}e^{-x}+\frac{2}{3}e^{3x}-2e^{x}-2xe^{-x}$

[Rui Xiang]

Here is my solution.

[BJM]

Here is the solution.

[Yuying Chen]

$\text{(a)}\\$
$r^2-2r-3=0\\$
$(r+1)(r-3)=0\\$
$r=-1, r=3\\$
$\text{Homogeneous Equation: $y_c(x)=c_1e^{-x}+c_2e^{3x}$}\\ \\$

$\text{Since we know $\cosh{x}=\frac{e^{x}+e^{-x}}{2} = \frac{1}{2}e^x+\frac{1}{2}e^{-x}$}\\$
$\text{Therefore,}\\$
$y^{\prime\prime}-2y^{\prime}-3y=8e^x+8e^{-x}\\$
$Y(x)=Ae^x+Bxe^{-x}\\$
$Y^{\prime}(x)=Ae^x+Be^{-x}-Bxe^{-x}\\$
$Y^{\prime\prime}(x)=Ae^x-Be^{-x}-Be^{-x}+Bxe^{-x}\\ $
$Ae^x-2Be^{-x}+Bxe^{-x}-2Ae^x-2Be^{-x}+2Bxe^{-x}-3Ae^x-3Bxe^{-x} = 8e^x+8e^{-x}\\$
\begin{cases}
  (A-2A-3A)e^{x}=8e^{x}\\\
  (-2B-2B)e^{-x}=8e^{-x}
   \end{cases}
\begin{cases}
  A=-2\\
 B=-2
   \end{cases}
$Y(x)=-2e^x-2xe^{-x}\\$
$\text{General Solution:}\\$
$y(t)=c_1e^{-x}+c_2e^{3x}-2e^x-2xe^{-x}\\$


$\text{(b)}\\$
$y^{\prime}=-c_1e^{-x}+3c_2e^{3x}-2e^{x}-2e^{-x}+2xe^{-x}\\$
$y(0)=0\Rightarrow c_1+c_2-2=0\\$
$y^{\prime}(0)=0\Rightarrow -c_1+3c_2-2-2=0\\$
\begin{cases}
  c_1=\frac{1}{2}\\
 c_2=\frac{3}{2}
   \end{cases}

$\text{Thus,}$
$y(t)=\frac{1}{2}e^{-x}+\frac{3}{2}e^{3x}-2e^x-2xe^{-x}$

[Xinqiao Li]

a) Find the general solution for equation $y'' - 2y' -3y = 16coshx$
b) Find solution, satisfying $y(0) = 0, y'(0) = 0$

a)
Consider $y'' - 2y' -3y = 0$

Assume $y = e^{rx}$, then the characteristic polynomial of the equation is given by $r^2 -2r -3 = 0$

This simplifies to $(r - 3)(r +1) = 0$ which gives us two roots $r_1 = 3$ and $r_2 = -1$

Then the complementary solution is $y_c = c_1e^{3x} + c_2e^{-x}$

Since $y'' - 2y' -3y = 16cosh = 8e^x + 8e^{-x}$

We guess the particular solution is of the form

$y = Ae^x + Bxe^{-x}$
$y' = Ae^x + B(e^{-x} - xe^{-x}) = Ae^x + Be^{-x} - Bxe^{-x}$
$y'' = Ae^x - Be^{-x} - B(e^{-x} - xe^{-x}) = Ae^x - 2Be^{-x} + Bxe^{-x}$

Substituting back to the original equation: 

$y'' - 2y' -3y = Ae^x - 2Be^{-x} + Bxe^{-x} - 2Ae^x - 2Be^{-x} + 2Bxe^{-x} - 3Ae^x - 3Bxe^{-x} = -4Ae^x - 4Be^{-x} = 8e^x + 8e^{-x}$

Therefore, $A = -2$ and $B = -2$
The particular solution is $y_p = - 2e^x - 2xe^{-x}$
The general solution is :
$$y = c_1e^{3x} + c_2e^{-x} - 2e^x - 2xe^{-x}$$

b)
$$y = c_1e^{3x} + c_2e^{-x} - 2e^x - 2xe^{-x}$$
$$y' = 3c_1e^{3x} - c_2e^{-x} - 2e^x + 2xe^{-x} - 2e^{-x}$$

Plug in the initial conditions $y(0) = 0, y'(0) = 0$, we have,
$0 = c_1 + c_2 - 2$
$0 = 3c_1 - c_2 - 2 - 2$

so, $c_1 = \frac{3}{2}$ and $c_2 = \frac{1}{2}$
The particular solution is
$$y = \frac{3}{2}e^{3x} + \frac{1}{2}e^{-x} - 2e^x - 2xe^{-x}$$

[yueyangyu]

a)
First solve the homogenous part:
\begin{aligned}
r^{2}-2r-3 &=0\\
r^{2}-2r+1&=4\\
(r-1)^{2} &=4\\
r_1 &=3 \\r_2 &=-1
\end{aligned}
So the solution to homogenous part is:
\begin{aligned}
y_c(x) =c_1e^{3x}+c_2e^{-x}
\end{aligned}
Next we solve
\begin{aligned}
y^{\prime\prime}-2y^{\prime}-3y &=16cosh(x)\\
                                                     &=16*\frac{e^{x}+e^{-x}}{2} \\
                                                     &=8e^{x}+8e^{-x}
\end{aligned}
Let
\begin{aligned}
y_p(x)&=Ae^{x}+Be^{-x}*x\\
y_p^{\prime}(x) &= Ae^{x}+Be^{-x}-Bxe^{-x}\\
y_p^{\prime\prime}(x) &= Ae^{x}-Be^{-x}-Be^{-x}+Bxe^{-x}\\
                                   &= Ae^{x}-2Be^{-x}+Bxe^{-x}\\
\end{aligned}
Thus we can have
\begin{aligned}
Ae^{x}-2Be^{-x}+Bxe^{-x}-2(Ae^{x}+Be^{-x}-Bxe^{-x})-3(Ae^{x}+Be^{-x}*x) &=8e^{x}+8e^{-x}\\
-4Ae^{x}-4Be^{x} &=8e^{x}+8e^{-x}\\
A &=-2\\
B &=-2
\end{aligned}
Therefore, we can have:
\begin{aligned}
y_p(x) &=-2e^{x}-2xe^{-x}
\end{aligned}
From the above, we get
\begin{aligned}
y&=y_c(x)+y_p(x)\\
&= c_1e^{3x}+c_2e^{-x}-2e^{x}-2xe^{-x}
\end{aligned}


b)
\begin{aligned}
y &= c_1e^{3x}+c_2e^{-x}-2e^{x}-2xe^{-x}\\
y_p^{\prime} &= 3c_1e^{3x}-c_2e^{-x}-2e^x+2xe^{-x}-2e^{-x}\\
\end{aligned}
since
\begin{aligned}
y(0) &=0\\
y^{\prime} &= 0
\end{aligned}
we can have
\begin{aligned}
c_1 &=3/2\\
c_2 &=1/2
\end{aligned}
Thus, the solution is
\begin{aligned}
y &= 3/2e^{3x}+1/2e^{-x}-2e^{x}-2xe^{-x}
\end{aligned}