# [SOLVED]Order and if eq is linear

#### karush

##### Well-known member
$\displaystyle x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+2y=\sin(x)$
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....

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#### HallsofIvy

##### Well-known member
MHB Math Helper
$\displaystyle x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+2y=\sin(x)$
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 text book 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

##### Well-known member
Check your text book 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

##### Well-known member
MHB Math Helper
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

##### Well-known member
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

##### Well-known member
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