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Sorry for my English .... here is my question

Professor gives me homework it's I have a verity of differential equation and i need to describe it with 4 thing

  1. What the order of the equation
  2. What is the degree of the equation
  3. If it's liner equation or not
  4. If it's a homogeneous equation or not

I want to know how to find these Help me with easy English... no use hard words

Ted Shifrin
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user430638
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1 Answers1

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  1. $n$-th order means the equation has derivatives of order $n$ at most. For example a second order differential equation would be $$ y''(x)+y(x)^5x=0.$$
  2. The degree is the largest exponent of a derivative. For example a differential equation of degree two would be $$ (y'''(x))^2+x=0.$$
  3. Look up what linear means. Basically linear differential equations have this form $$ y^{(n)}(x)+a_{n-1}y^{(n-1)}(x)+\dots+a_2y''(x)+a_1y'(x)+a_0y(x)=b$$ where $a_0,\dots,a_n,b$ are elements of a field (constants) and $n\in \mathbb{N}$. These linear equations have the beautiful property that any linear combination of solutions is a solution to the equation again! So say you solve the equation and find that the function $f$ and the function $g$ are solutions, then you know that the sum $f+g$ is also a solution.
  4. For a linear differential equation this means that $b=0$ and if you want to define it for non linear equations, see this answer.
ty.
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    I disagree with your answer to 4. – Ted Shifrin Sep 27 '17 at 20:03
  • @TedShifrin Indeed you are correct. I had a very diffuse understanding of homogeneous up until 1 minute ago. :D I will try to make a correct statement on 4. – ty. Sep 27 '17 at 20:16
  • You have to specify how the equation is homogeneous. For instance, $y''y + y'^2-x y^2=0$ is homogeneous under rescaling of $y$. $x y''-y'=7/x$ is homogeneous under rescaling of $x$. $y'' + y' /x +y^2 /x^3=0$ is homogeneous under the simultaneous rescaling of $x$ and $y$. – user121049 Sep 27 '17 at 21:53
  • I disagree with 2. The order of that ODE would be 3, not 2. The order of an ODE is the highest order derivative of the unknown function appearing in the ODE, and is not affected by algebraic operations on those derivatives. e.g., $y' + 3xy = 0$, $(y')^{277}y + 3x\cos(y) = 0$, and $e^{xy'}+y = 0$ have the same order, 1, even though the last two are certainly more complicated than the first. – Nicholas Stull Oct 03 '17 at 14:56
  • @NicholasStull oops, I meant degree, not order. I will edit my answer. – ty. Oct 03 '17 at 18:13