Sometimes it arrives to me that I try to solve a linear differential equation for a long time and in the end it turn out that it is not homogeneous in the first place.
Is there a way to see directly that a differential equation is not homogeneous?
Please, do tell me.
For a linear differential equation $$a_n(x)\frac{d^ny}{dx^n}+a_{n-1}(x)\frac{d^{n-1}y}{dx^{n-1}}+\cdots+a_1(x)\frac{dy}{dx}+a_0(x)y=g(x),$$ we say that it is homogenous if and only if $g(x)\equiv 0$. You can write down many examples of linear differential equations to check if they are homogenous or not. For example, $y''\sin x+y\cos x=y'$ is homogenous, but $y''\sin x+y\tan x+x=0$ is not and so on. As long as you can write the linear differential equation in the above form, you can tell what $g(x)$ is, and you will be able to tell whether it is homogenous or not.
The simplest test of homogeneity, and definition at the same time, not only for differential equations, is the following:
An equation is homogeneous if whenever $\varphi$ is a solution and $\lambda$ scalar, then $\lambda\varphi$ is a solution as well.
A homogeneous differential equation have same power of $X$ and $Y$ example :$- x+y dy/dx= 2y$
$X+y$ have power $1$ and $2y$ have power $1$ so it is an homogeneous equation.
The best and the simplest test for checking the homogeneity of a differential equation is as follows :--> The formulae is $$\frac{dy}{dx}=F(x,y)$$ such that $$F(x,y)=F(tx,ty)$$ for "t" being any arbitrary constant, then $$\frac{dy}{dx}$$ is homogeneous.
Take for example we have to solve $$\frac{dy}{dx}=\frac{y+ (x^2+y^2)^\frac12}{x}$$
Put $x=tx$ and $y=ty$ where t is any arbitrary constant.
Now from the numerator and denominator take the constant as common with maximum power possible from both numerator and denominator each.
If the constant gets cancelled throughout and we obtain the same equation again then that particular differential equation is homogeneous and the the power of constant which remains after cutting it to lowest degree is the degree of homogeneity of that equation. Hope That helped!
Longer answer: let f(x) = |x|. f(x) is a splitwise function, where f(x) = x for x>=0 and f(x) = -x for x<0. incidentally f(a) = f(-a), but that does not mean that a = -a.
– Patrick vD Sep 26 '24 at 22:51if you are given an ODE say $f(x,y)=x^2-3xy+5y^2$ and they ask you to show if it is homogeneous or not here is how to do it
If a function $f$ has the property that $f(tx,ty)=t^nf(x,y)$ then the function is homogeneous of degree n...to prove if the above DE is homogeneous here is how to do it \begin{align*} f(x,y)&=x^2-3xy+5y^2\\ f(tx,ty)&=(tx)^2-3(tx)(ty)+5(ty)^2\\ &=t^2x^2-3t^2xy+5t^2y^2\text{ if we factor $t^2$ we get}\\ &=t^2[x^2-3xy+5y^2]\\ &=t^2f(x,y) \end{align*} hence the function is homogeneous of degree $n$
Any function like y and its derivatives are found in the DE then this equation is homgenous
ex. y"+5y´+6y=0 is a homgenous DE equation
But y"+xy+x´=0 is a non homogenous equation becouse of the X funtion is not a function in Y or in its derivatives
I think a differential equation is homogeneous if every term contains y or derivatives of y in the equation
The equations in the form $f(xy)$ can be said to be homogeneous also if they can be put in the form $dy/dx =f(y/x)$ or in other cases $f(x,y )=x^n g(y/x)$
