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.

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How to tell if a differential equation is homogeneous, or inhomogeneous?

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?…
VVV
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Find answer of $\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$

If I had a very shallow question, then I am sorry. $x,y,z\in\mathbb{N}^{+}$ and$$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}=4$$find $x,y,z$. I try with AM-GM, just get$$ \frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{y+x}\geq\frac{3}{2}$$ This means that the…
Zuo
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Can a differential equation be non-linear and homogeneous at the same time?

I have searched for the definition of homogeneous differential equation. I have found definitions of linear homogeneous differential equation. Can a differential equation be non-linear and homogeneous at the same time? (If yes then) what is the…
Siddhartha
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What is the motivation behind a product solution?

Let's consider the simple differential equation: $$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$ And let's assume we have some regular homogeneous boundary conditions like: $$ u(a, y) = 0$$ $$ u(L, y) = 0$$ $$ u'(x,…
turnip
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Why can't I solve this homogenous second order differential equation?

I've been banging my head on the wall for quite some time trying to come up with a solution to the following: $$\frac {\partial^2 y(x)} {\partial x^2} + (A-B*V(x)) y(x) = 0 $$ $$V(x) = (36 + (2 - x)^2)^{-1/2}$$ With A and B constants, and $y$ solely…
TreyE
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Evaluating $\int_{0}^{\infty} \frac{\mathrm{d}x}{x(I_n(x)^2 + K_n(x)^2)}$ and similar integrals.

This question is inspired by this evaluation of an integral by Michael Penn. Suppose we have a differential equation $y''+p(x) y' +q(x)y =0$ with linearly independent solutions $y_1(x)$ and $y_2(x)$ over some interval $I$. Given that $p,q$ are…
Robert Lee
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Solving non homogeneous recurrence relation

I am having a hard time understanding these questions. I know I need to find the associated homogeneous recurrence relation first, then its characteristic equation. I cant figure out how to find the particular solution to the non homo recurrence…
htcSpt
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Multivariate polynomial functional equation

I’m having some difficulties solving the following functional equation: Find all polynomials $P(x,y)\in\mathbb{R}[X,Y]$ for which: $P(x,y)$ is homogeneous (so $\exists n\in\mathbb{N}, \forall x,y,t\in\mathbb{R}: P(tx,ty)=t^n\cdot…
Jonas De Schouwer
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Solve the differential equation $\frac{dy}{dx}=\frac{y+2y^5}{4x+y^4}$

Solve the differential equation $$\frac{dy}{dx}=\frac{y+2y^5}{4x+y^4}$$ My try: we can write the equation as: $$\frac{dy}{dx}=\frac{1}{y^3}\frac{\left(1+2y^4\right)}{1+\frac{4x}{y^4}}$$ Multiplying both sides with $\frac{1}{y^5}$ we…
Umesh shankar
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Fourier transform of $\frac {x} {(x^2+y^2)}$

I was trying to compute the Fourier transform of $f(x,y)=\frac{x}{x^2+y^2}$. I saw in a paper I was reading that $$\hspace{4cm}\hat{f}(\xi_1,\xi_2)=Const.\frac{\xi_1}{\xi_1^2+\xi_2^2}\hspace{4cm} (*) $$ (i.e. $f$ works like a eigenvetor for the…
user609149
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Can I turn $Ax=b$ into $Ax=0$?

For a system of equations $$ \begin{bmatrix}d_1 & d_2 & \dots & d_n \end{bmatrix} \begin{bmatrix}u_1\\u_2\\ \vdots \\ u_n \end{bmatrix} = d_{n+1} $$ where each $d$ is a column of (possibly noisy) data and each $u$ is a scalar unknown, is the correct…
user171848
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Second order inhomogeneous equation: $y''-2xy'-11y=e^{-ax}$

My question relates to an a second order inhomogeneous equation: $$y''-2xy'-11y=e^{-ax}$$ First I need to investigate the homogeneous equation: $$y''-2xy'-11y=0$$ $$y''-2xy'=11y$$ Forms Hermite's Equation where $\lambda = 11$ So I need a general…
Student146
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What is the physical interpretation of homogeneous?

In ode's and pde's we pay great attention as to whether the equations are homogeneous or nonhomogeneous. I remember learning in my first ODE class that for the general linear…
Zduff
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How many rational solutions does $(x^2+4x+8)^2+3x(x^2+4x+8)+2x^2$ have?

How many rational solutions does $(x^2+4x+8)^2+3x(x^2+4x+8)+2x^2$ have? I don't know how to start...
joko34
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Stability of limit cycle associated with a homogenous linear quation

Study the stability of the limit cycle r=1 for the system given in polar coordinates by the equations $\dot{r}=(r^2−1)(2x−1), \dot{\phi}=1$, where $x=r\cos \phi$. I've been trying to solve this problem by estimating the return function, but…
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