-b.1.3.1 Order and if eq is linear

[karush]

2000
$\displaystyle
x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+2y=\sin(x)
$
I probably am not advanced enough to understand this but thot I would take a shot at it

the order of this is second due to the order of the highest derivative that appears.
but I didn't see why this is a linear equation..

The book defines this "The differential equation

$\displaystyle F\left(x,y',y''...y^n\right)=0$

is said to be linear if \(\displaystyle F\) is a linear function of the variables \(\displaystyle x,y',y''...y^n\)

thanks for any help on this...

 

[MarkFL]

I have moved this thread to our Differential Equations subforum.

A linear ODE of order $n$ with variable coefficients has the general form:

\(\displaystyle \sum_{k=0}^n\left[p_k(x)\frac{d^ky}{dx^k} \right]=Q(x)\)

Linear differential equation - Wikipedia, the free encyclopedia

 

[HallsofIvy]

$\displaystyle
x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+2y=\sin(x)
$
I probably am not advanced enough to understand this but thot I would take a shot at it

the order of this is second due to the order of the highest derivative that appears.
but I didn't see why this is a linear equation..

The book defines this "The differential equation

$\displaystyle F\left(x,y',y''...y^n\right)=0$

is said to be linear if \(\displaystyle F\) is a linear function of the variables \(\displaystyle x,y',y''...y^n\)

Check your textbook again. I would be very surprised if it said this because it isn't true. What is true is that the equation $$F(x, y', y'', ..., y^(n))= 0$$ is said to be linear if F is a linear function of $$y', y'', ..., y^(n)$$. Do you see the difference? F does not have to be linear in the independent variable, x.

You can write $$x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+2y=\sin(x)$$ as
$$x^2y''+ xy+ 2y= sin(x)$$ where the only non-linear functions are of x: $$x^2$$ and $$sin(x)$$.

thanks for any help on this...

 

[karush]

Check your textbook again. I would be very surprised if it said this because it isn't true. What is true is that the equation $$F(x, y', y'', ..., y^(n))= 0$$ is said to be linear if F is a linear function of $$y', y'', ..., y^(n)$$. Do you see the difference? F does not have to be linear in the independent variable, x.

View attachment 2087

scanned from the book "Elementary Differential Equations and Boundary Value Problems"
I guess there is a difference...
So I did read the Wiki on this ... so I presume a linear eq when plotted is a straight line..

what would be an example of the eq

$\displaystyle x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+2y=\sin(x) $ since it is a linear eq but has \(\displaystyle x^2\) and \(\displaystyle \sin(x)\) in it

 

[HallsofIvy]

Okay, so you do see now that what you wrote before is not what is said in your book.

You said before that

The differential equation $$F(x, y, y', y'', ..., y^n)= 0$$ is said to be linear if F is linear function of x, y, y', ..., $$y^n$$.

What you post now, from your book, says $$F(x, y, y', y'', ..., y^{(n)})= 0$$ is said to be linear if F is a linear function of y, y', ..., $$y^{(n)}$$.

The difference is that x is NOT included in the list after "F is a linear function of". F may be a non-linear function of x but still give a linear differential equation as long as it is a linear function of the dependent variable, y, and its derivatives.

 

[karush]

OK that helps I will try the other problems
I would hit the thanks button but it doesn't appear on my mobil phone

 

[karush]

I need to continue with this but have to move to another subject so will mark this as solved. this reply's were certainly helpful