Toronto Math Forum

4) $x'=\left(\begin{array}{cc}
3 & 3\\
-2 & -1
\end{array}\right)x$

a)

$det(A-\lambda I)=det(\begin{array}{cc}
3-\lambda & 3\\
-2 & -1-\lambda
\end{array})=(3-\lambda)(-1-\lambda)+6=0$

$\lambda^{2}-2\lambda+3=0$

$(\lambda-1)^{2}=-2$

$\lambda=1\pm\sqrt{2}i$

$\lambda=1+\sqrt{2}i$

$(\begin{array}{ccc}
2-\sqrt{2}i & 3 & 0\\
-2 & -2-\sqrt{2}i & 0
\end{array}) = (\begin{array}{ccc}
2-\sqrt{2}i & 3 & 0\\
0 & 0 & 0
\end{array}) $

$v=(\begin{array}{c}2+\sqrt{2}i\\-2\end{array})$

$e^{(1+\sqrt{2}i)t}(\begin{array}{c}
2+\sqrt{2}i\\
-2
\end{array})=e^{t}(cos\sqrt{2}t+isin\sqrt{2}t)(\begin{array}{c}2+\sqrt{2}i\\-2\end{array}) = e^{t}(\begin{array}{c}
2cos\sqrt{2}t+2isin\sqrt{2}t+\sqrt{2}icos\sqrt{2}t-\sqrt{2}sin\sqrt{2}t\\
-2cos\sqrt{2}t-2isin\sqrt{2}t
\end{array})=e^{t}(\begin{array}{c}
2cos\sqrt{2}t-\sqrt{2}sin\sqrt{2}t\\
-2cos\sqrt{2}t
\end{array})+ie^{t}(\begin{array}{c}
2isin\sqrt{2}t+\sqrt{2}icos\sqrt{2}t\\
-2isin\sqrt{2}t
\end{array})$

$x(t)=c_{1}e{}^{t}(\begin{array}{c}
2cos\sqrt{2}t-\sqrt{2}sin\sqrt{2}t\\
-2cos\sqrt{2}t
\end{array})+c_{2}e{}^{t}(\begin{array}{c}
2isin\sqrt{2}t+\sqrt{2}icos\sqrt{2}t\\
-2isin\sqrt{2}t
\end{array})$

$$
M_{y}=1+6ye^{3x}
$$
$$
N_{x}=4ye^{2x}    
$$
$M_{y} ≠N_{x} $,it is not exact
$$

R_{2} =\frac{ M_{y} -N_{x}}{N}=\frac{1+2ye^{2x}  }{1+2ye^{2x}}=1
$$
$$
μ=e^{∫R_{2}dx} =e^{∫1 dx} =e^x   
$$
Multiplying both sides by $\mu$, we get
$$

ye^x+3y^2e^{3x} +(e^x+2ye^{3x}) y^\prime=0
$$
$$
M_{y}^\prime=e^x+6ye^{3x}
$$
$$
N_{x}^\prime=e^x+6ye^{3x}
$$
$M_{y}^\prime=N_{x}^\prime$,it is exact
$$

∃φ(x,y) such that\ φ_{x} =M^\prime,φ_{y} =N^\prime
$$
$$
φ(x,y)=∫{M^\prime dx}=∫{ye^x+3y^{2}e^{3x}dx}=ye^x+y^{2}e^{3x} +h(y)
$$
$$
φ_{y} =e^x+2ye^{3x} +h(y)^\prime=e^x+2ye^{3x}
$$
$h(y)^\prime=0$
$$
$$
So h(y)=c
$$
φ(x,y)=ye^x+y^{2}e^{3x} =c
$$
Since y(0)=1
$$
1⋅e^0+1^2⋅e^0=2=c
$$
$$
φ(x,y)=ye^x+y^{2}e^{3x} =2
$$

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}

Find the general solution of the given differential equation.
$$y''-2y'-2y=0$$

Assume that $$y=e^{rt}$$ and it follows that r must be a root of characteristic equation
$$r^2-2r-2=0$$
$$r=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
Then,
$$r_1=1+\sqrt{3} \quad r_2=1-\sqrt{3}$$

Therefore, the general solution of the given differential equation is:
$$y=c_1e^{(1+\sqrt{3})t}+c_2e^{(1-\sqrt{3})t}$$