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)

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Second order homogeneous linear differential equation with variables coefficients. [closed]

I'm facing an equation preventing me from moving forward in my studies. $$\frac{d^2 \delta}{dy} + \frac{2+3y}{2y(1+y)} \frac{d \delta}{dy} = \frac{3}{2} \frac{\delta}{y (1+y)}$$ I found both solutions ...
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Nonhomogeneous nonlinear differential equation with delta functions

I'm trying to solve the following differential equation $$ y'' + \dfrac{1}{2}(y')^2 = A \delta(x) + B \delta(x-a) + C $$ I tried two times, the first one using Laplace transforms, but I don't really ...
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Classification of Homogeneous functions

Is every homogeneous function of degree 1 in two variables is of the form $f(x,y)=\frac{p(x,y)}{q(x,y)}$, where $p(x,y)$ is a homogeneous polynomial of degree $n$ and $ q(x,y)$ is a homogeneous ...
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When and how do we dehomogenize a homogeneous function?

When and how do we dehomogenize a homogeneous function? To solve Prove the sign and zeroes of $Ax^2 + 2Bxy + Cy^2$ (without using the second derivative test) , "user" set $$t = \frac x y$$ ...
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General Solution for a non-homogenous ordinary differential equation

The differential equation is $\frac{d^2y}{dt^2} - \frac{dy}{dt} -6y = e^{3t} - 3t^2$. I first found the homogeneous solution which I got as $$y(t) = c_1e^{-2t} + c_2e^{3t}$$ I am trying to figure out ...
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Techniques for solving a difference recurrence relations

I’m having issues understanding how an answer is derived in a math textbook I have. Just looking for a technique for the derivation as well as some intuition to help with my understanding. The ...
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Homogenizing rational function in projective field to find poles gives different poles depending on which $X_i$ is used for homogenizing

Lets say we have a wierstrauss normal form elliptic curve $C : y^2 = x^3 + Ax^2 + B$. Then we look at the vertical line in the function field $f/g = (x - a)/1 \in K(C)$ which intersects at points $(a, ...
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Homogeneous linear differential equation of constant coefficients.

Determine all real valued solutions of equation : $(D^3 -i D^2+D-i)y = 0 $ where $ D=d/dx $ and $i$ is iota. The three roots are $i,±i$; and i don't how to proceed further. one of my doubt is that ...
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Prove that if $f(x, y)$ is homogeneous of degree 1, then $f_{xx} f_{yy} = (f_{xy})^2$

By definition of homogeneity, if $f(x, y)$ is homogeneous of degree $n$, then $$f(tx, ty) = t^nf(x, y)$$ Problem Prove that if $f(x, y)$ is homogeneous of degree $1$, then $f_{xx}(x, y) f_{yy}(x, y) = ...
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Finding a particular solution to undetermined coefficients problem that will be part of the fundamental set

The problem at hand My working so far: I am not sure where to go from here.
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Why do we need to keep the solution of the homogeneous equation in the general equation?

On the theme of differential equations, I wonder why we still need to keep the solution of the homogeneous equation. For example the linear differential equation : $y' + ay = x^2 \enspace \enspace \...
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Given $x>0,y>0$, AND $\frac{3}{2 x^2+3 xy}+\frac{5}{3 xy+4 y^2} =2$, So max{xy}?

Given $x>0;y>0$; if $\frac{3}{2 x^2+3 xy}+\frac{5}{3 xy+4 y^2} =2$, Find max{xy}? Here is my try: Solution 1: $xy=t$, $\frac{3}{2 x^2+3t}+\frac{5}{3t+4y^2} =2$ $y = \frac{t}{x}$ $\frac{3}{2 x^...
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How to find the summation of the following series? .

If $$S=\sum_{i=1}^{n} \frac{1}{i2^{i}},$$ Then how can I find the summation of the above series up to $n^{th}$ terms? I can't solve this question because I don't know whether this summation is a ...
Shivam Kumar's user avatar
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Given $a,b,c \in \mathbb{R}$, and $x,y,z \in \mathbb{R}$, with $x,y,z$ not all zero. Find $a^2+b^2+c^2+2abc$

This question is to be solved in about $3$ minutes, without a calculator. Let $a,b,c$ be any real numbers. Suppose that $x,y,z$ are real numbers, not all simultaneously zero, such that $$x=cy+bz$$ $$...
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The eigenvalues of a matrix with homogenous entries is also homogenous

Consider a matrix depending on a parameter $\xi$: $$ A(\xi)=\begin{pmatrix} a_{11}(\xi) & a_{12}(\xi) & \cdots & a_{1n}(\xi) \\ a_{21}(\xi) & a_{22}(\xi) & \cdots & a_{2n}(\xi) ...
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Why can we ignore the absolute value when solving certain homogeneous differential equations?

I have a question about the solution to the following homogeneous differential equation: $ydx=(x+\sqrt{xy})dy$. Specifically, I am struggling to understand the textbook's solution to it, which is: $$...
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Concern about the degree of rational functions that are Quasi-homogeneous in Differential Equations

I came across an interesting set of problems as follows: \begin{gather} \dfrac{dy}{dx} =\dfrac{1-xy^2}{2x^2y} \tag{1}\label{1} \\ \dfrac{dy}{dx}=\dfrac{2+3xy^2}{4x^2y} \tag{2}\label{2} \\ \dfrac{dy}{...
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Homogeneous ODE with multiple changes of variables

How to solve the homogeneous ODE $$ \frac{dx}{dt} = \frac{2tx}{t^2 - x^2} $$ I’ve changed the variable such that $y=x/t$ and I came up with the following $$ \frac{dy}{dt} = -\frac{y}{t} + \frac{2yt}{1-...
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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 ...
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Homogeneous or Nonhomogeneous ODEs?

May I ask whether the ODEs below are 'Homogeneous' or 'Nonhomogeneous'? 1.) y'' - y' = y 2.) y'' - y' = sin(y) 3.) y'' - y' = xy Thank you for your answers!
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Condition for lines joining origin to the point of intersection(s) of two curves to be perpendicular

Given curves: $$ax^2+2hxy+by^2+2gx=0$$ $$a_1x^2+2h_1xy+b_1y^2+2g_1x=0$$ It is given that the lines joining the origin to the points of intersection of these curves are perpendicular to each other. We'...
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How to find a solution to a homogeneous ordinary differential equation with non-constant coefficients (the coefficients are linear functions)?

I am trying to find a solution to a differential equation in the form $$y'' + (ax+b)y' + (cx+d)y = 0,$$ where $a,b,c,d$ are constants. I only know how to solve when the coefficients are constants, and ...
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Linear second order homogeneous matrix ODEs with constant coefficients: Solution strategies?

$\newcommand{\bm}[1]{\boldsymbol{#1}}$ $\newcommand{\img}{\operatorname{img}}$ The Scalar Setting When looking for a solution $u:\mathbb{R}\to\mathbb{R}$ of the following linear homogeneous second ...
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When does a homogeneous equation represent a set of straight lines?

Recently I came across this question, where the top voted answer claimed that all homogeneous equation represent a set of straiight lines passing through origin. I was wondering if this was true ...
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Solving a Lamé differential equation with a parameter out of boundaries.

I am trying to get rid of the following homogeneous ode. \begin{equation} \begin{split} u''(z)+\frac{1}{2} \left(\frac{1}{z}+\frac{1}{z-1}+\frac{1}{z-1}\right) u'(z)+\frac{2\left(A+B\right)- \left(2 ...
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On implicit equations of rational quadratic Bézier curves

Rational quadratic Bézier curve with control points $\boldsymbol{B}_0 = [x_0 : y_0: w_0], \boldsymbol{B}_1 = [x_1 : y_1: w_1], \boldsymbol{B}_2 = [x_2 : y_2: w_2]$ in homogeneous coordinates of $\...
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Particular Integral of a Homogenous Linear PDE

Consider, $z$ a function of $z= f(x,y)$ and D & D' respectively represent partial differential operators, i.e $$\frac{∂}{∂𝑥}, \quad\frac{∂}{∂y}$$ i am learning particular integrals for PDE using ...
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Physical significance of repetitive roots of a PDE

Consider, the following Linear PDEs with homogenous coefficients, $$(D^4 + D'^4 -2D^2D'^2)z =0; \qquad......{\rm eqn}.1$$ $$(D^3D'^2 + D^2D'^3)z =0; \qquad......{\rm eqn}.2$$ Here z is a ...
Banoffee π's user avatar
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In cases of one multiple root (differential equation characteristic polynomial). How do we know that $xe^{rx}$ is another solution?

Given a differential equation, say: $$y''+2y'+y=0$$ we might get two equal roots to the characteristic polynomial. Here we get $r_1=r_2=-1$. So, $e^{-x}$ is a solution. In order to get another ...
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Solving homogeneous differential equation $y''(x) = ky(x)$ using different approaches leads to different results

I have the following second order homogeneous linear differential equation: $$y''(x) = k y\left( x \right), \quad\quad k \in \mathbb{R}$$ Using the exponential approach $$y\left( x \right) = d e^{r x}...
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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 $y_2(x)$ ...
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Solving the 1D Heat Equation on [a,b] rather than [0,L]

Solve the 1D Heat Equation on $x \in [a,b]$ $$ \frac{\partial ^2 T}{\partial x^2} = \frac{1}{\alpha} \frac{\partial T}{\partial t}$$ $$ T(a,t) = T(b,t) = 0, T(x,0) = T_0(x) $$ Now, I know that ...
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Imposing boundary conditions on third order differential equation's solution

I am called to solve the following differential equation $$F'''(r)+\frac{\partial\log R(r)}{\partial r}F''(r)+ \frac{\partial^2\log R(r)}{\partial r^2} F'(r)=0$$ with $r\in(-\infty,0]$. The solution, ...
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Solution of equation $(1 - c)[(2 - c) ^ 2 - 9] = 0$

I was solving a characteristic equation for matrix eigenvalue problem. I had this equation, where $c$ is eigenvalue. $$(1 - c)[(2 - c) ^ 2 - 9] = 0$$ In the book they equated both the terms to zero, ...
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Solving linear nonhomogeneous differential equation

I have to solve such an equation: \begin{equation} y^{\prime \prime}+\frac{\pi}{3}y^{\prime}+ay=b \end{equation} where $a$ and $b$ are constants. I can`t pick up a partial solution, so I hope for your ...
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how to caculate the step response of the second order system with nonzero initial condition, $y¨ + 2ξω_ny˙ +2y = 0, y(0) = 0, y˙(0) = c$

How to caculate the step response of the second order system with nonzero initial condition,$$ \ddot{y}+ 2ξω_n\dot y +2y = 0, y(0) = 0, \dot y(0) = c$$ (where $c$ is a contant) Question: how to ...
user1923435's user avatar
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Prove that the tangent line is given by $\frac{\partial F}{\partial X}(P)X+\frac{\partial F}{\partial Y}(P)Y+\frac{\partial F}{\partial Z}(P)Z=0$

This is problem A.$5$ in Rational Points on Elliptic Curves: Let $C: F(X, Y, Z) = 0$ be a projective curve given by a homogeneous polynomial $F \in \mathbb C[X, Y, Z]$, and let $P \in \mathbb P^2$ be ...
Clyde Kertzer's user avatar
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Homogeneous equations for conic sections

The equation of a "standard" circle is $x^2 + y^2 = r^2$. That equation is not homogeneous and does not include the origin; we can homogenize it by adding a $z$ term of degree 2, getting $x^...
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How to integrate $xy' = \sqrt{x^2-y^2}+y$

I have to integrate: $xy' = \sqrt{x^2-y^2}+y$ It is supposed to be an homogeneous differential equation, but I can't see the variable change to make it happens. I tried: $y' = \sqrt{1-(y/x)^2} + y/x$ ...
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Solution of homogeneous differential equation stays above zero solution

Let $I \subseteq R$ be an interval such that $0 \in I$ and $f \in C^1(\mathbb{R},\mathbb{R})$ with $f(0) = 0$. For some $t_0 \in I, y_0 \in \mathbb{R}$, consider the initial value problem $$ \begin{...
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Tangent cone of affine line at the origin

I am trying to calculate the tangent cone of $\mathbb{A}^1$ at the point $Q=1$. To do that, I first want to calculate the tangent cone at the origin, and then "translate back". Here is the ...
EJB's user avatar
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Example of tangent cone of affine variety

I am reading "Lectures on Étale Cohomology" by J.S. Milne and I am trying to understand example 2.7. It says that the tangent cone of the affine variety defined by $Y^2=X^3+X^2$ at the ...
EJB's user avatar
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Solve $\left(3x^2y-xy\right)dx+\left(2x^3y^2+x^3y^4\right)dy=0$

$\left(3x^2y-xy\right)dx+\left(2x^3y^2+x^3y^4\right)dy=0$ I'm trying to solve this first-order differential equation. I know it's not an exact equation so I'm trying to use the method taught in class ...
eddie's user avatar
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Non-separable linear PDE with separable solutions

Consider the PDE $c^{2} y_{xx} = y_{tt} + 2 \gamma y_{t}$ (this is a wave equation with damping). If $\gamma$ is spatially varying, and so dependent on $x$, I can't see a clear way to separate the PDE....
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Is $(0,...,0) \in \mathbb{R}^n$ the unique critical point of the linear system of constant coefficients?

If I've got the following differential linear homogenous system of constant coefficients $$x_1'(t)=a_{11}x_1+a_{12}x_2+...+a_{1n}x_n $$ $$x_2'(t)=a_{21}x_1+a_{22}x_2+...+a_{2n}x_n$$ $$... $$ $$x_n'(t)=...
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What does Ax=0 has only the trivial solution imply? [duplicate]

There is a true or false question that asks if A is a n x n matrix and if Ax=0 has only the trivial solution, then the system Ax = b has a unique solution for every b that is real. I believe that this ...
testcase0_'s user avatar
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2 answers
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Deducing parameters of 2-by-2 linear system of first-order ODEs from data

I'm currently researching glucose-insulin regulation system and I came across this homogeneous system of linear differential equations: \begin{array}{lr} x'&=&-\alpha x-\beta y\\ y'&=&\...
Woojin Rho's user avatar
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Matrices for which the systems of linear equations $Ax= 0$ and $Bx= 0$ have the same solutions.

Let $n,m≥1$, let $\Bbb K$ be a field and let $A,B\in \Bbb K^{m×n}$ be matrices for which the systems of linear equations $Ax= 0$ and $Bx= 0$ have the same solutions. (A) $C\in \Bbb K^{n×n}$ exists ...
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Second order recurrence relation when initial term is defined for $n$ different from $0$ (characteristic root method)

Let's say the initial conditions of a second order recurrence relation are given for $v_{-1}$ and $v_{0}$ rather than the classical $v_{0}$ and $u_{1}$. It seems that the usual formula for the ...
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Solve differential equation equal to constant

I am having a confusing moment. This should be simple, but I am still stumbling around an inconsistency. Below the differential equation in question: $$\frac{dT_\Delta}{dt}=cT_\Delta$$ where $T_\Delta ...
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