Understanding Linearity of Differential Equations

[APUGYael]

Hey all,

I don't understand what makes a differential equation (DE) linear.
I found this: "x y' = 1 is non-linear because y' is not multiplied by a constant"
but then also this: "x' + (t^2)x = 0 is linear in x".

t^2 also isn't a constant.
So why is this equation linear?

 

[Orodruin]

A differential operator $\hat L$ is linear if $\hat L (a f_1 + b f_2) = a\hat L f_1 + b \hat L f_2$, where a and b are constants and the fs functions.

A differential equation for a function $f$ is linear if it can be written as $\hat L f = g$, where g is some fixed function that does not depend on f. If g=0 the differential equation is homogeneous.

x y’’(x) = 1 is an inhomogeneous linear differential equation. Your second example is a homogeneous linear differential equation.

 

[APUGYael]

A differential operator $\hat L$ is linear if $\hat L (a f_1 + b f_2) = a\hat L f_1 + b \hat L f_2$, where a and b are constants and the fs functions.

A differential equation for a function $f$ is linear if it can be written as $\hat L f = g$, where g is some fixed function that does not depend on f. If g=0 the differential equation is homogeneous.

x y’’(x) = 1 is an inhomogeneous linear differential equation. Your second example is a homogeneous linear differential equation.

I don't understand, still. What makes y(t) different from t². They're both non-constants.
If you define y(t)=t^2 then you would be able to do y(t) (a+b) = ay(t)+by(t) = at^2 +bt^2, right?

 

[Orodruin]

You do not "define" $y(t)$, you need to solve the differential equation to find out what $y(t)$ is. It has nothing to do with whether or not the coefficient in front of $y$ is constant in the independent variable or not. The only thing that matters is the property I described above, i.e., whether or not the differential equation can be written $\hat L y = g$ for some linear differential operator $\hat L$ or not. To take your first example, your differential operator would be $\hat L = x (d/dx)$ and $g = 1$, leading to
$$
\hat L y(x) = x\frac{dy}{dx} = x y'(x) = 1.
$$
The operator $\hat L$ is linear because
$$
\hat L [a_1 y_1(x) + a_2 y_2(x)] = x \frac{d}{dx}[a_1 y_1(x) + a_2 y_2(x)] = a_1 x \frac{dy_1}{dx} + a_2 x \frac{dy_2}{dx} = a_1 \hat L y_1 + a_2 \hat L y_2.
$$

Edit: Effectively, the DE is linear if it is a sum of terms where each term contains only one factor of $y(x)$ or its derivatives. It does not matter whether the coefficients are constant in the independent variable or not.

Edit 2: So the general linear ODE of order $n$ is of the form
$$
\sum_{k = 0}^n f_k(x) \frac{d^ky}{dx^k} = g(x).
$$
The ODE is homogeneous if $g(x) = 0$ and otherwise inhomogeneous.

 

[APUGYael]

You do not "define" $y(t)$, you need to solve the differential equation to find out what $y(t)$ is. It has nothing to do with whether or not the coefficient in front of $y$ is constant in the independent variable or not. The only thing that matters is the property I described above, i.e., whether or not the differential equation can be written $\hat L y = g$ for some linear differential operator $\hat L$ or not. To take your first example, your differential operator would be $\hat L = x (d/dx)$ and $g = 1$, leading to
$$
\hat L y(x) = x\frac{dy}{dx} = x y'(x) = 1.
$$
The operator $\hat L$ is linear because
$$
\hat L [a_1 y_1(x) + a_2 y_2(x)] = x \frac{d}{dx}[a_1 y_1(x) + a_2 y_2(x)] = a_1 x \frac{dy_1}{dx} + a_2 x \frac{dy_2}{dx} = a_1 \hat L y_1 + a_2 \hat L y_2.
$$

Edit: Effectively, the DE is linear if it is a sum of terms where each term contains only one factor of $y(x)$ or its derivatives. It does not matter whether the coefficients are constant in the independent variable or not.

Edit 2: So the general linear ODE of order $n$ is of the form
$$
\sum_{k = 0}^n f_k(x) \frac{d^ky}{dx^k} = g(x).
$$
The ODE is homogeneous if $g(x) = 0$ and otherwise inhomogeneous.

Can you show me an example of a non-linear equation and can you show me (using the operator) why it isn't linear, please?

 

[Orodruin]

An example of a non-linear differential equation would be
$$
y'(x) + y(x)^2 = 0.
$$
It is non-linear because
$$
[y_1(x)+y_2(x)]' + [y_1(x)+y_2(x)]^2 \neq [y_1'(x) + y_1(x)^2] + [y_2'(x) + y_2(x)^2].
$$

 

[APUGYael]

An example of a non-linear differential equation would be
$$
y'(x) + y(x)^2 = 0.
$$
It is non-linear because
$$
[y_1(x)+y_2(x)]' + [y_1(x)+y_2(x)]^2 \neq [y_1'(x) + y_1(x)^2] + [y_2'(x) + y_2(x)^2].
$$

Why am I so confused...?

 

[Orodruin]

Your differential operator cannot depend on the dependent function itself. Furthermore, you cannot write
$$
y'(x) + y(x)^2 = y(x) [d/dx + y(x)],
$$
the derivative needs to act on $y(x)$. You could write it as
$$
[d/dx + y(x)]y(x),
$$
but it is still not linear because the first parenthesis depends on $y(x)$.

 

[APUGYael]

Your differential operator cannot depend on the dependent function itself. Furthermore, you cannot write
$$
y'(x) + y(x)^2 = y(x) [d/dx + y(x)],
$$
the derivative needs to act on $y(x)$. You could write it as
$$
[d/dx + y(x)]y(x),
$$
but it is still not linear because the first parenthesis depends on $y(x)$.

I am sorry. I don't understand any of this.Hello,

First of all my apologies. Turns out I never had the subject of operators yet. After watching a video with an explanation I think it is now totally clear on what a linear operator is defined as and how to apply an operator to a function.

I see now what you were saying in the beginning. I might've been doing alternative math all this time, which is never a good idea because it's fictional.

Thanks for the help ;-)

-Yael