integrating dy = x^3 dx - 3x^2 dx + x dx to get y = (x^4)/4 - x^3 + (x^2)/2 + c

[bobdvt]

Hello,

At 49 years of age I'm catching up on the math I didn't care to understand earlier on in life.

I'm reading "Differential Equations Workbook for Dummies" by Steven Holzner.

On page 10 (for those who have the book), I stuck on an elliptic explanation as to integrating an equation.

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Time for a more advanced problem!

. . . . .\(\displaystyle \dfrac{dy}{dx}\, =\, x^3\, -\, 3x^2\, +\, x, \quad y(0)\, =\, 3\)

Because this equation doesn't involve any terms in y, you can move the dx to the right like this:

. . . . .\(\displaystyle \color{green}{\mbox{1. }\, dy\, =\, x^3\ dx\, -\, 3x^2\ dx\, +\, x\ dx}\)

Then just integrate to get

. . . . .\(\displaystyle \color{green}{\mbox{2. }\, y\, =\, \dfrac{x^4}{4}\, -\, x^3\, +\, \dfrac{x^2}{2}\, +\, c}\)

To evaluate c, use the initial condition, which is

. . . . .\(\displaystyle y(0)\, =\, 3\)

Plugging \(\displaystyle x\, =\, 0\) and \(\displaystyle y(0)\, =\, 3,\) into the equation for y gives you

. . . . .\(\displaystyle y(0)\, =\, 3\, =\, c\)

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Could someone please explain how you get from 1 to 2?

Thank you!

 

[tkhunny]

At 49 years of age I'm catching up on the math I didn't care to understand earlier on in life.

I'm reading "Differential Equations Workbook for Dummies" by Steven Holzner.

On page 10 (for those who have the book), I'm stuck on an elliptic explanation as to integrating an equation.

---

Time for a more advanced problem!

. . . . .\(\displaystyle \dfrac{dy}{dx}\, =\, x^3\, -\, 3x^2\, +\, x, \quad y(0)\, =\, 3\)

Because this equation doesn't involve any terms in y, you can move the dx to the right like this:

. . . . .\(\displaystyle \color{green}{\mbox{1. }\, dy\, =\, x^3\ dx\, -\, 3x^2\ dx\, +\, x\ dx}\)

Then just integrate to get

. . . . .\(\displaystyle \color{green}{\mbox{2. }\, y\, =\, \dfrac{x^4}{4}\, -\, x^3\, +\, \dfrac{x^2}{2}\, +\, c}\)

To evaluate c, use the initial condition, which is

. . . . .\(\displaystyle y(0)\, =\, 3\)

Plugging \(\displaystyle x\, =\, 0\) and \(\displaystyle y(0)\, =\, 3,\) into the equation for y gives you

. . . . .\(\displaystyle y(0)\, =\, 3\, =\, c\)

---

Could someone please explain how you get from 1 to 2?

It's just a linearity property. Just draw an integral sign in front of EACH term.

 

[bobdvt]

Thank you tkhunny for your reply.

It's just a "linearity property". Just draw an "integral sign" in front of EACH term

... Well, I just don't get it.

Initially, I had a problem with the elliptic explanation in the book. Now after reading your reply and investigating both "linearity property" and "integral sign", I'm stuck in the same place:

-Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line
-The symbol

is used to denote an integral

.The symbol was invented by Leibniz and chosen to be a stylized script"S" to stand for "sum"

I'm still puzzled as to how one gets from 1) to 2)...

Cheers!

 

[mmm4444bot]

I'm still puzzled as to how one gets from 1) to 2) …

This is not a question about differential equations. It's a question about calculus.

Have you studied calculus, yet?


When integrating polynomials, we integrate term-by-term (i.e., integrate each term in the polynomial separately), using the Power Rule from integral calculus.

The integral of the term x^3 is x^4/4

The integral of the term -3x^2 is -x^3

The integral of the term x is x^2/2

Symbol c is the constant-of-integration term, and it's very important to not forget to include it after integrating.

If it seems confusing, then you ought to begin by studying calculus first.

PS: I've proofread two different "for dummies" math books. They were crap, rife with errors in logic, typographical mistakes, mismatched graphics, arithmetic mistakes, and other stuff that was just plain wrong. I'm not familiar with the book that you referenced; however, I would suggest a bonafide textbook or web site for any serious student of calculus. :cool:

 

[bobdvt]

Thank you mmm4444bot for your reply!

"Polynomial"... "power rule"..."integral calculus"

Those got me on the right track.

Cheers!