# Diff eq with constants... Eulers identity...

In summary, the person is trying to solve a second order differential equation, but they are not understanding why their answer is different from what they found in a book. They use the correct identity to get their answer, but their constants are not the same as what is in the book.

## Homework Statement

Find the general solution of the second order DE.

$y'' + 9y = 0$

## The Attempt at a Solution

Problem is straight forward I just don't get why my answer is different than the books.

So you get

$m^2 + 9 = 0$
$m = 3i$ and $m = -3i$

so the general solution would be:

$c_1e^{3ix} + c_2e^{-3ix} = y$

my book gives me

$e^{i\theta} = cos(\theta) + isin(\theta)$

from there I get

$e^{iβx} = cos(βx) + isin(βx)$
$e^{i-βx} = cos(βx) - isin(βx)$

I have

$e^{i3x} = cos(3x) + isin(3x)$
$e^{i-3x} = cos(3x) - isin(3x)$

so I get $y = c_1cos(3x) + c_1isin(3x) + c_2cos(3x) -c_2isin(3x)$

but my book gives me

$y = c_1cos(3x) + c_2sin(3x)$

I feel like my answer is still valid for some reason.. I just don't know how they got their answer from my answer. I used the correct identity..

Your ##c_1## and ##c_2## are just not the same constants as those of the book.

Orodruin said:
Your ##c_1## and ##c_2## are just not the same constants as those of the book.

So I guess that means my answer is still valid then... I just don't get what they used for c1 and c2... did they use i or something? because I notice there is no i term in their answer..

## Homework Statement

Find the general solution of the second order DE.

$y'' + 9y = 0$

## The Attempt at a Solution

Problem is straight forward I just don't get why my answer is different than the books.

So you get

$m^2 + 9 = 0$
$m = 3i$ and $m = -3i$

so the general solution would be:

$c_1e^{3ix} + c_2e^{-3ix} = y$

my book gives me

$e^{i\theta} = cos(\theta) + isin(\theta)$

from there I get

$e^{iβx} = cos(βx) + isin(βx)$
$e^{i-βx} = cos(βx) - isin(βx)$

I have

$e^{i3x} = cos(3x) + isin(3x)$
$e^{i-3x} = cos(3x) - isin(3x)$

so I get $y = c_1cos(3x) + c_1isin(3x) + c_2cos(3x) -c_2isin(3x)$

but my book gives me

$y = c_1cos(3x) + c_2sin(3x)$

I feel like my answer is still valid for some reason.. I just don't know how they got their answer from my answer. I used the correct identity..

Just write ##a = c_1 + c_2## and ##b = i c_1 - i c_2##. Your solution becomes ##a \cos 3x + b \sin 3x## for two constants ##a## and ##b##.

BTW: when using LaTeX, do NOT write ##cos(\theta)##, etc.; write, instead, ##\cos( \theta)## or ##\cos \theta##, which you get by typing "\cos" instead of "cos", The notation ##cos \theta## looks ugly and is hard to read, but ##\cos \theta## looks good and is clear.

Gotcha. Ty everyone!

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It describes the relationship between a variable and its rate of change.

## 2. What are constants in a differential equation?

Constants in a differential equation are coefficients that remain constant throughout the equation. They do not depend on the independent variable and help to define the behavior of the equation.

## 3. What is Euler's identity?

Euler's identity is a mathematical formula that relates five important numbers: 0, 1, pi, e, and i (the imaginary unit). It is written as e^(i*pi) + 1 = 0 and is considered one of the most beautiful and profound equations in mathematics.

## 4. How is Euler's identity used in differential equations?

Euler's identity is often used in differential equations to solve problems involving complex numbers and exponential functions. It can also be used to simplify and transform differential equations into more manageable forms.

## 5. What is the significance of Euler's identity in science and engineering?

Euler's identity has many important applications in science and engineering, particularly in fields such as physics, electrical engineering, and signal processing. It is also closely related to the principles of calculus, making it a fundamental concept in many areas of mathematics.

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