4
Assume that a Lyapunov function $V$ exists then
$$\dot{V} = \Big(x\dfrac{\partial V}{\partial x}-y^{3}\dfrac{\partial V}{\partial y}\Big)(y^4-2x^2)-y\dfrac{\partial V}{\partial x}+2x\dfrac{\partial V}{\partial y}$$
This suggests the function $V = x^2+\dfrac{y^2}{2}$ which is postive definite and radially unbounded. Further, we have
$$\dot{V} = -(y^4-2x^2)^2$...
3
Using $x_1 = x$ and $x_2 = y$, you have the nonlinear planar system
$$
\begin{align}
\dot{x}_1 &= x_1 x_2^4 - 2 x_1^3 - x_2 \\
\dot{x}_2 &= 2 x_1 + 2 x_1^2 x_2^3 - x_2^7
\end{align} \tag{1}
$$
which has an equilibirum at $(x_1, x_2) = (0, 0)$. We can choose the Lyapunov function
$$
V(x_1, x_2) = x_1^2 + \frac{1}{2} x_2^2
$$
which is globally ...
1
Apologies, I would of preferred to try my best to finish the problem, but I'm just so busy. It's hard for me to picture the limit does not hold for a large class of well behaved functions. On the other hand surely we can contrive functions which will break the limit. By work such as Lebesgue's I feel finding a necessary and sufficient condition on functions ...
1
Once you have $P(t)=Ce^{rt}$, the statement that it doubles in five years means $\frac {P(5)}{P(0)}=2$. The value of $C$ does not matter because it divides out. In fact, $C=P(0)$. It doesn't matter whether $P(0)=1$ or $1000$ or something else. Now you have $$2=e^{5r}\\ \log 2=5r\\r=\frac {\log 2}5$$ which is not the same as $r=\sqrt[5]2$. $\sqrt [5]2$ ...
1
If you take $\alpha=4$ you get the directional derivative equal zero. That means that $V$ is not just some Lyapunov function but it is a first integral of your system. The orbits are level sets of such $V$ which are circles. $(0,0)$ is stable but not asymptotically stable.
1
Hint: Consider the general equation:
$$
m\ddot{x}+c\dot{x}+kx =F_0(t)\tag1
$$
and let $x(\omega,t)$ be the particular solution of:
$$m\ddot{x}+c\dot{x}+kx =e^{i\omega t}.\tag2$$
Then the particular solution of $(1)$ can be computed as:
$$
x(t) = \int_{-\infty}^\infty F_0(\omega)x(\omega,t)d\omega,
$$
where
$$
F_0(\omega)=\int_{-\infty}^\infty F_0(t)e^{-i\...
1
Trying to write a short and easy to read answer:
The amount of substance in water: $$x*V$$
The change rate of substance in water: $$(x*V)'$$
The change rate of water level: $$V' = s_1-s_2$$
The rate at which the substance is flowing into the tank is: $$0.1*s_1$$
The rate at which the substance is flowing out of the tank is: $$x*s_2$$
So you end up with ...
1
You need to find the rate of change of alcohol in the tank.
The rate is foun by rate in minus rate out
Rate in is $$0.10s_1$$
Rate out is $$\frac {x(t)}{V(t)}s_2$$
Thus
$$x'(t)=0.10s_1-\frac {x(t)}{V(t)}s_2$$
Assuming that the volume stays constant the equation is easier to solve.
1
You are right that $v_1 = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$ is an eigenvector of $A$ with eigenvalue $2$; i.e $ v_1 \in \ker(A-2I)$. But like you mentioned, the dimension of this kernel is $2$, so you need to find another linearly independent eigenvector. It is easy to verify that $\xi = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} $ is such a vector. ...
1
Your eigenvectors are not right for $\lambda=2.$ You have the right system, but you should get two solutions: $v_1=(1,1,0)$ and $v_2=(1,0,1).$
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