Simple initial value problems - pros and cons of different methods

Question

Consider the problem:

Find $f(x)$ if $f’(x)=4x$ and $f(3)=12$

I have always done this, and taught it, as a two-step problem: First, find the general anti-derivative, $f(x)=2x^2+C$, and then plug in the given value, $12=2(3)^2+C$, to solve for $C$.

I recently saw a student do it in one step: $$f(x)=12+\int_3^x 4t \,dt$$

I had not seen this before, and I find it kind of brilliant but kind of bizarre. Do many people do such problems this way? I like its directness, but it also seems a bit less accessible, somehow. Does anyone have experience with teaching these problems both ways? How do students do with these two methods? What are the respective advantages and pitfalls?

Thanks in advance for any insights.

Answer 1

Differential equations are employed to construct models of physical phenomena. In particular, consider the very simple linear non-autonomous differential equation

$$y'(x) = g(x)$$

with initial condition $y(x_0) = y_0$. Linear differential equations of this type usually describe a physical system forced by the term $g(x)$ which starts from a certain initial state.

Your two-step solution does not give a formal solution valid for any integrable $g(x)$ and any initial condition $y_0$. It seems that any $g(x)$ and any $y_0$ should be treated in a special way, and there's no way to speak of the general properties of a solution.

The one-step solution is instead a particular case of

$$y(x) = \int_{x_0}^x g(t) \,\mathrm{d}t + y(x_0),$$

a form which is valid for any integrable $g(x)$ and $y_0$. But this, in turn, is a particular case of the convolution integral

$$y(x) = \int_{x_0}^x h(x-t)g(t) \,\mathrm{d}t + y(x_0),$$

which represents the response of any general linear time-invariant system — represented by $h(x)$ — with one input forced by the term $g(x)$.

These forms allow to discuss general properties of the solutions (e.g. natural and forced response, eigenfunctions and eigenvalues, frequency response, stability), something that it would not be possible by treating each differential equation as a special case. Of course, one should not give the impression that any differential equation possesses solutions that can be written explicitly in this way, but when this is possible there are many advantages.

The solution that you find bizarre is actually quite common in every application of ordinary differential equations.

Answer 2

Look in your textbook where the Fundamental Theorem of Calculus is discussed. You might see the formula $$f(x)\; =\; f(a) + \int_a^x f'(t) \,dt$$ or a very close version of this, such as $$f(x) - f(a) \; = \;\int_a^x f'(t) \,dt.$$

I personally have not seen a student do this, but I can easily imagine a student who wasn't paying attention in class solving this by chancing upon these general equations, which are usually displayed quite prominently in texts. You might want to ask this student how he/she came up with this method, and maybe share it with the class, with credit being given to this student.

As for what to do in the future, I recommend that you continue to teach it as you have been, which is pretty much universal now and for at least the last 150 years (based on my extensive experience in looking at old textbooks), because it is less formularic and it is the method used in many other contexts --- solving differential equations, curve-fitting, definitions by recursion, etc. in which step 1 is to obtain a general solution (or identify the general form a solution takes) and step 2 is to solve for the specific constants that are involved in the general solution.