What Are Key Concepts in Gerald Teschl's Section 1.2 on ODEs?

[Rasalhague]

I've started reading Gerald Teschl's Ordinary Differential Equations and Dynamical Systems. I'd desperately like to crack these definitions from section 1.2: classical ODE, linear, homogeneous, first order system, autonomous, as they seem pretty fundamental. My plan is to give answers to the excercises at the 1.3-1.8. I'd be grateful if anyone could comment on any aspect of my answers, offer corrections, clear up confusion, or suggest further resources for self-study. If this post is too long and involved to respond to particular points, any confusion-quelling hint or pointer relating to the terminology would be useful.


Problem 1.3. Classify the following differential equations. Is the equation linear, autonomous? What is its order?

(i) y'(x) + y(x) = 0. Answer: It's a 1st order classical ODE, since it takes the form of equation (1.12), at the beginning of section 1.2, namely F(t,x,x⁽¹⁾,...x^(k)) = 0, rather than equating the highest derivative with some expression involving the independent variable, the dependent variable (unknown function) and less high derivatives of the dependent variable. It's nonlinear, since it doesn't take the form of equation (1.17). It's not homogeneous, since homogeneity has only been defined for linear ODEs. It's not autonomous because it doesn't take the form of the system (1.19).

Alternative answers: 1st order, linear, homogeneous and autonomous, since it can be made so by rearranging it as y'(x) = -y(x), which doesn't "depend on x", and is therefore equivalent to such an equation. Or perhaps it's not autonomous, since we don't know whether y is of the form (t,x,x',...,x^(k)). So I guess maybe it just depends on what x is, whether it represents the function Teschl calls the dependent variable, as I suspect it does, or a tuple of functions of the independent variable, like z in (1.19).


(ii) u''(t) = t sin(u(t)). Answer: 2nd order, nonlinear (since it's not a linear combination of u and derivatives of u with coefficient functions of t) and therefore not homogeneous; not autonomous, since t appears on the right.

Alternative answer: linear, homogeneous, not autonomous, taking the function on the right as g in (1.17)--since it is, after all, a function of t--and letting all the f(t)'s in (1.17) be identically equal to 0.


(iii) y(t)² + 2y(t) = 0. Answer: Unclassified. It resembles classical, but is more restricted, in that, while it takes the form y(x) + y'(x) = 0, it isn't the most general expression of this form. Question: is this even a differential equation? Is there a name for such an equation, where the derivatives are implicit?

(iv) D²₁u(x,y) + D²₂u(x,y) = 0, where superscripts on D denote second derivative, and subscript 1 and 2 denote partials with respect to the 1st and 2nd argument respectively. Answer: PDE. A "classical" PDE? (With the 0 on one side, it seems analogous to a classical ODE.)

(v) x' = -y; y' = x. Answer: Unclassified?


Problem 1.4. Which of the following differential equations for y(x) are linear?

(i) y' = sin(x)y + cos(y).
(ii) y' = sin(y)x + cos(x).
(iii) y' = sin(x)y + cos(x).

Answer, assuming y' means y'(x): (iii). (It's not homogeneous, because the cos(x) term is not multiplied by y. It's not autonomous, since x appears explicitly? No, I'm not at all sure about that last statement.)


Problem 1.5. What is the most general form of a 2nd order linear equation?

Answer:

$$x_i''(t) = g_i(t) + \sum_{j=1}^{n} \left [ f_{ij}(t) \cdot x_j(t)+h_{ij}(t) \cdot x_j'(t) \right ]$$


Problem 1.6. Transform the following differential equations into first-order systems.

(i) x'' + t sin(x') = x. Answer: y = (x,x'); y' = (x',x'') = (x',x-t sin(x')). I'm guessing that's a better form to express the answer in than (arcsin[(x-x'')/t],x-t sin(x')).

(ii) x'' = -y, y'' = x. Answer: a = (x,x'); a' = (x',x'') = (y''',-y)? (Does this count as a differential equation? It seems to be a very different entity from the previous example, in that it contains another, non-first order system, in terms of a completely different unknown function, embedded inside it. Is there a name for this? Or does y represent a known function?)

Aside. This suggests that order, at least, is not inherent in the relation, but depends on the form it's written in, and that rearranging an equation is considered to turn it into a different equation, perhaps with a different order. I wonder if the same rule applies to other terms, such as linear, homogeneous, autonomous; I've assumed it does, which is why I've relegated the other possibility to "anternative answers". I wonder though if some rearrangements are considered trivial enough not to count as changing the equation, such as adding a number to both sides.


Problem 1.7. Transform the following differential equations into autonomous
fi rst-order systems.

(i) x'' = t sin(x') = x. Answer: z = (t,x,x'); z' = (1,x',x'') = (1,x',t sin(x')).

(ii) x'' = - cos(t)x. Answer: z = (t,x,x'); z' = (1,x',x'') = (1,x',- cos(t)x).

The last equation is linear. Is the corresponding autonomous system also
linear? Answer: Yes.


Problem 1.8. Let x^(k) = f(x; x⁽¹⁾,...,x^(k-1)) be an autonomous equation (or system). Show that if phi(t) is a solution, then so is phi(t - t₀).

Answer: I don't know. Perhaps I need to go stare at the implicit function theorem some more. I wonder if it has anything to do with the issue mathwonk spoke of in this thread. But if it's "a crucial property in differential equations, namely when is the solution defined for all 'time'", there must sometimes be cases where the claim in Problem 1.8 can't be taken for granted. Why would autonomousness matter, I wonder, given that any system can be expressed in autonomous form, presumably without changing the problem. Or is the mention of autonomy a red herring?

 

[jvicens]

Thank you for sharing your thoughts and questions on Gerald Teschl's Ordinary Differential Equations and Dynamical Systems. I am glad to see that you are eager to understand the fundamental concepts in section 1.2 and are actively engaging with the exercises in section 1.3-1.8. I am a scientist with a background in mathematics and I would be happy to offer some feedback and suggestions on your answers and questions.

Firstly, let me clarify some of the definitions from section 1.2 that seem to be causing confusion. A classical ODE is a differential equation that can be written in the form F(t,x,x',...,x(k)) = 0, where t is the independent variable, x is the dependent variable (unknown function), and x',...,x(k) are the derivatives of x with respect to t. This is in contrast to implicit equations, where the derivatives are not explicitly present, as in the case of (iii) in Problem 1.3. In this case, y(t)2 + 2y(t) = 0, it is not a differential equation because there are no derivatives present. It is just an algebraic equation, and there is no specific name for it.

A linear differential equation is one that can be written in the form y(k) + a(k-1)y(k-1) + ... + a(0)y = g(t), where y is the dependent variable, t is the independent variable, and a(k-1),...,a(0) and g(t) are functions of t. This means that the equation is a linear combination of the dependent variable and its derivatives, with coefficient functions of t. In (i) of Problem 1.4, the equation is linear because it can be written as y' = sin(x)y + cos(y), which is in the form mentioned above. In (ii) and (iii), the equations are not linear because they cannot be written in this form.

A homogeneous differential equation is one where the right-hand side (g(t) in the linear equation above) is identically equal to 0. This means that the equation does not depend on t, and only involves the dependent variable and its derivatives. In (i) of Problem 1.4, the equation is not homogeneous because the right-hand side is not 0. In (ii) and (iii), the equations are homogeneous because the right-hand