# 微分方程学代写|DIFFERENTIAL EQUATIONS MATH221 University of Liverpool Assignment

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## Instructions:

Differential equations are indeed fundamental to many areas of mathematics and science, including physics, engineering, economics, and biology, just to name a few. They allow us to model and analyze complex systems and phenomena that are too difficult or impossible to understand using algebraic or geometric techniques alone.

The five parts of the module you mentioned seem to cover a broad range of topics in differential equations, from basic first-order ODEs to more advanced PDEs. It’s great that the module emphasizes both theory and applications, as both are important for understanding and using differential equations effectively.

Solving differential equations can be a difficult and sometimes frustrating task, as there are often many different methods and techniques that can be used, and the solutions can be complex and difficult to interpret. However, with practice and patience, it’s possible to become proficient in solving and analyzing these equations.

Overall, it sounds like MATH201 will be a valuable and challenging module for students who are interested in pursuing further studies in mathematics, science, or engineering. Good luck to all who take it!

In (a)-(c) we consider the autonomous equation $\dot{x}=2 x-3 x^2+x^3$.
(a) Sketch the phase line of this equation.

(a) The equation is $\dot{x}=x(x-1)(x-2)$. The phase line has three equilibria $x=0,1,2$. For $x<0$, the arrow points down. For $02$, the arrow points up.

(b) Sketch the graphs of some solutions. Be sure to include at least one solution with values in each interval above, below, and between the critical points.

(b) The horizontal axis is $t$ and the vertical axis is $x$. There are three constant solutions $x(t) \equiv 0,1,2$. Their graphs are horizontal. Below $x=0$, all solutions are decreasisng and they tend to $-\infty$.

Between $x=0$ and $x=1$, all solutions are increasing and they approach $x=1$. Between $x=1$ and $x=2$, all solutions are decreasing and they approach $x=1$. Above $x=2$, all solutions are increasing and they tend to $+\infty$.

(c) Some solutions have points of inflection. What are the possible values of $x(a)$ if a nonconstant solution $x(t)$ has a point of inflection at $t=a$ ?

(c) A point of inflection $(a, x(a))$ is where $\ddot{x}$ changes sign. In particular, $\ddot{x}(a)$ must be zero. Differentiating the given equation with respect to $t$, we have
$$\ddot{x}=2 \dot{x}-6 x \dot{x}+3 x^2 \dot{x}=\dot{x}\left(2-6 x+3 x^2\right) \text {. }$$
If $x(t)$ is not a constant solution, $\dot{x}(a) \neq 0$ so that $x(a)$ must satisfy
$$2-6 x(a)+3 x(a)^2=0 \quad \Leftrightarrow \quad x(a)=1 \pm \frac{1}{\sqrt{3}} .$$