Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [homogeneous-equation]

A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi(x)$ is a solution, so is $c \phi(x)$, where c is an arbitrary (non-zero) constant. (Def: http://en.m.wikipedia.org/wiki/Homogeneous_differential_equation)

A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi(x)$ is a solution, so is $c \phi(x)$, where $c$ is an arbitrary (non-zero) constant. Reference: Wikipedia.

Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable $y$ must contain $y$ or any derivative of $y$. A linear differential equation that fails this condition is called inhomogeneous.

584 questions
0
votes
1 answer

Defining Homogeneous Differential Equations

I am putting together a list of types of first and second order differential equations and I am struggling with the definition of homogeneous and nonhomogeneous. Can anyone clarify the definitions for me please?
Matt Bartlett
  • 1
0
votes
0 answers

Why an homogeneous inequality can be rewritten as another one?

Consider the following homogeneous inequality $$ A^{T}*P+PA\lt0, P > 0$$ where $A$ is a square given matrix. Since the inequality above is homogeneous, it can be rewriten as $$A^{T}*P+PA<-I, P \geq I$$ Why is this possible? Thanks
marianauc
  • 1
0
votes
0 answers

Linear non-homogeneous recurrent relation

A linear nonhomogeneous recurrence relation of degree k with constant coefficients is a recurrence relation of the form: $ a_n= c_1 a_{n-1}+c_2 a_{n-2}+\ldots+c_ka_{n-k}+f(n)$ where $c_1,c_2,\ldots,c_k$ are real numbers and $f(n)$ is a function not…
T.Nguyen
  • 111
0
votes
3 answers

Proof that smooth positive degree $m$ homogeneous function is polynomial of degree $m$ and $m$ is a positive integer

So I know that by Euler's homogeneous function theorem $m$ is a positive number, but why is it an integer? And how to prove that $f$ is polynomial of degree $m$?
Bar Dubovski
  • 89
0
votes
1 answer

How to solve this equation by converting it from non-homogenous form to homogenous form and then telescoping?

This equation below must solved by converting it from a non-homogenous equation into a homogenous one and then use the characteristic equation. T(1)= 5 T(n)= 2T(n−1)+3n+1,n>1 In the text and pictures that follow, I show how I solved and verified…
mrcoder123
  • 1
0
votes
2 answers

transpose of System of equations

Let $A$ be an $m\times n$ matrix of rank $n$ with real entries. Choose the correct statements. 1. $Ax=b$ has a solution for any $b$. 2. $Ax=0$ does not have a solution. 3. If $Ax=b$ has a solution, then it is unique. 4. $y'A=0$ for some non zero…
Khuraijam Shitle Kumar
  • 1
0
votes
2 answers

find general equation of $x''(t) + 5x'(t) + 4x(t) = 0$

Suppose $x_1(t)$ and $x_2(t)$ are two linearly independent solutions of the equations: $$x'_1(t) = 3x_1(t) + 2x_2(t)$$ and $$x'_2(t) = x_1(t) + 2x_2(t)$$ where $x'_1(t)\text{ and }x'_2(t)$ denote the first derivative of functions $x_1(t)$ and…
ARNAB2012
  • 1
-1
votes
1 answer

Degree in homogeneous function in differential equations

How do we say $n$ to be degree of an equation, We have $F(kx,ky)=k^{n} F(x,y)$ then we say n is the degree of the equation but we generally consider the degree to be the highest power of a variable in a polynomial but here the $k$ is an arbitrary…
Razz
  • 133
1
2