Meeting Mr. Bernoulli

September 30, 2018August 3, 2021 / Michael Xue / 3 Comments

The differential equation

${d \over dx} y + f(x) y = g(x) y^{\alpha}\quad\quad\quad(1)$

where $\alpha \neq 0, 1$ and $g(x) \not \equiv 0$ , is known as the Bernoulli’s equation.

When $\alpha$ is an integer, (1) has trivial solution $y(x) \equiv 0$ .

To obtain nontrivial solution, we divide each term of (1) by $y^{\alpha}$ to get,

$\boxed{y^{-\alpha}{d \over dx}y} + f(x) y^{1-\alpha} = g(x)\quad\quad\quad(2)$

Since ${d \over {dx}}({{1 \over {1-\alpha}}y^{1-\alpha}}) ={1 \over {1-\alpha}}\cdot (1-\alpha) y^{1-\alpha-1}{d \over dx}y=\boxed{y^{-\alpha}{d \over dx}y}$ ,

(2) can be expressed as

${d \over dx} ({{1 \over {1-\alpha}} y^{1-\alpha}}) + f(x) y^{1-\alpha} = g(x)$

which is

${{1 \over {1-\alpha}} {d \over dx} y^{1-\alpha}} + f(x) y^{1-\alpha} = g(x)$ .

Multiply $1-\alpha$ throughout,

${d \over dx} y^{1-\alpha} + (1-\alpha) f(x) y^{1-\alpha} = (1-\alpha) g(x)\quad\quad\quad(3)$

Let $z = y^{1-\alpha}$ , (3) is transformed to a first order linear equation

${d \over dx} z + (1-\alpha) f(x) z = (1-\alpha) g(x)$ ,

giving the general solution of a Bernoulli’s equation (see Fig. 1)

Screen Shot 2018-09-28 at 4.36.05 PM.png

Fig. 1

For examples of Bernoulli’s equation, see “What moves fast, will slow down” and “An Epilogue to ‘A Relentless Pursuit’“.

Pandora’s Box

September 24, 2018December 26, 2018 / Michael Xue / 2 Comments

Summations arise regularly in mathematical analysis. For example,

$\sum\limits_{i=1}^{n}{1 \over {i (i+1)}} = {n \over {n+1}}$

Having a simple closed form expression such as ${n \over {n+1}}$ makes the summation easier to understand and evaluate.

The summation we focus on in this post is

$\sum\limits_{i=1}^{n}i 2^i\quad\quad\quad(1)$

We will find a closed form for it.

In a recent post, I derived the closed form of a simpler summation (see “Beer theorems and their proofs“) Namely,

$\sum\limits_{i=0}^{n}x^i={{x^{n+1}-1} \over {x-1}}\quad\quad\quad(2)$

From (2) it follows that

${d \over {dx}}{\sum\limits_{i=0}^{n}x^i} = {d \over {dx}}({ {x^{n+1}-1} \over {x-1} })$

which gives us

${\sum\limits_{i=0}^{n}{{d \over dx}x^i}}={{(n+1)x^{n}(x-1)-(x^{n+1}-1)} \over {(x-1)^2}}$ .

Or,

${\sum\limits_{i=0}^{n}{i x^{i-1}}}= {{\sum\limits_{i=0}^{n}{i x^{i}}} \over {x}}$

$= {{(n+1)x^{n}(x-1)-(x^{n+1}-1)} \over {(x-1)^2}}$ .

Therefore,

${\sum\limits_{i=1}^{n}{i x^{i}}}={{(n+1)x^{n+1}(x-1)-x^{n+2}+x} \over {(x-1)^2}}$ .

Let $x=2,$ we arrived at (1)’s closed form:

${\sum\limits_{i=1}^{n}i 2^i} = {{(n+1)2^{n+1} -2 ^{n+2} + 2} \over {2-1}} = 2^{n+1} (n-1) + 2$ .

I have a Computer Algebra aided solution too.

Let $s_n \triangleq \sum\limits_{i=1}^{n} i x^i$ ,

we have $s_1 = x, s_{n}-s_{n-1}=n x^n$

Therefore, the closed form of $s_n$ is the solution of initial-value problem

$\begin{cases} {s_{n}-s_{n-1} }= {n x^n} \\ s_1=x\end{cases}$

It is solved by Omega CAS Explorer (see Fig. 1)

Screen Shot 2018-09-21 at 3.24.33 PM.png

Fig. 1

At ACA 2017 in Jerusalem, I gave a talk on “Generating Power Summation Formulas using a Computer Algebra System“.

I had a dream that night. In the dream, I was taking a test.

It reads:

Derive the closed form for

$\sum\limits_{i=1}^{n} {1 \over {(3i-2)(3i+1)}}$

$\sum\limits_{i=1}^{n} {1 \over {(2i+1)^2-1}}$

$\sum\limits_{i=1}^{n} {i \over {(4i^2-1)^2}}$

$\sum\limits_{i=1}^{n} {{i^2 4^i} \over {(i+1)(i+2)}}$

$\sum\limits_{i=1}^{n} { i \cdot i!}$

I woke up with a sweat.

My shot at Harmonic Series

September 5, 2018September 5, 2018 / Michael Xue / 1 Comment

Screen Shot 2018-08-29 at 12.40.22 AM.png

To prove Beer Theorem 2 (see “Beer theorems and their proofs“) is to show that the Harmonic Series $1 + {1 \over 2} + {1 \over 3} + ...$ diverges.

Below is my shot at it.

Yaser S. Abu-Mostafa proved a theorem in an article titled “A differentiation test for absolute convergence” (see Mathematics Magazine 57(4), 228-231)

His theorem states that

Let $f$ be a real function such that ${d^2 f} \over {dx^2}$ exists at $x = 0$ . Then $\sum\limits_{n=1}^{\infty} f({1 \over n})$ converges absolutely if and only if $f(0) = f'(0)=0$ .