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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When an m by n matrix A has rank r = m, the system Ax = b can be solved for which b(best answer)? How many special solutions to Ax = O?

I've been struggling with this question from Linear Algebra - Step by Step by KULDEEP SINGH, Oxford University Press. When an m by n matrix A has rank r = m, the system Ax = b can be solved for which ...
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Basis of solutions of $x^{2}y''-4xy'+6y=0$?

What is the basis of solution space spanned by the solutions of the homogeneous ODE given by $$x^{2}y''-4xy'+6y=0$$ My work: The solution is of the type $x^{m}$. By solving the auxiliary equation for ...
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Do elementary column operations change the null space

I know elementary row operations do not change the null space or the row space but they do change the column space. Likewise, elementary column operations change the row space but not the column space....
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Find the difference equation given the general solution $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$

Given that $y(k) = c_{1}5^{k} + c_{2}(-5)^{k} + c_{3}6^{k}$, is the general solution to a difference equation, how do you work backwards to find the difference equation?
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how to find the general soultion using reduction of order?

I'm trying to find the general solution using reduction of order for this ode $$y′′ + 2y′ + y = e^{-x}$$ i have found the complementary soultion : $$c_1e^{-x}+c_2e^{-x}x$$ but im unsure how to use ...
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How can Floquet be used to solve periodic linear equations?

I generally understand how to solve linear systems using x' = Ax where x is a vector, and A is a matrix. However, I am lost when the A matrix becomes periodic. I am ...
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Solving a Fourth-Order Linear Homogeneous Differential Equation

I like to solve the ordinary fourth order homogeneous differential equation given by $\displaystyle \frac{d^{4}\theta}{d z^{4}} + \lambda \cdot \theta = 0$ with a constant coefficient $\lambda$. ...
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How to use the solution of an homogeneous non linear DE when plugging a discretized input.

there's a certain class of non-linear (NL) DE (non linear in $f$) $$ x' = f(x)+g(t), $$ the homogeneous form of which admits an analytic solution. Suppose the solution to such an homogeneous form $x'=...
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How to tell if PDE boundary conditions are homogeneous or not?

Say we have Laplace's equation: $$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$ These are the boundary conditions. Now, to solve this by separation of variables, we need ...
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Limit of homogeneous ODE solutions when degeneracy is recovered

If I have a second order homogenous ODE $$y''+(4+a)y'+(4+2a)y = 0 $$ for a constant $a$, the solutions come from the roots of the polynomial $x^2+(4+a)x+4+2a=0$, which are $x=-2$ and $x=-2-a$. If $a\...
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Differential Equations: example problem from Morris Tenebaum's and Henry Pollard's book

I was working through the book 'Ordinary Differential Equations' by Morris Tenebaum and Henry Pollard. I came across the following differential equation: $$ (\frac{-vy}{\sqrt{x^2 + y^2}}+ wcos(\alpha))...
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Proving $f=\alpha g$ for some $\alpha\in\mathbb{R}$ if $f,g$ are homogeneous functions of degree $m,n$ respectively and $h=f+g$, $xh_{x}+yh_{y}=0$

I am supposed to prove that if $f(x,y)$ and $g(x,y)$ are homogeneous functions of degree $m$ and $n$ respectively and that $h=f+g$, such that $xh_{x}+yh_{y}=0$, then for some scalar $\alpha\in\mathbb{...
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Solving homogenous linear PDE: $(D^{2}+2DD'-D'^{2})z=\cos(3x+y)$

The question to solve the homogenous linear PDE: $(D^{2}+2DD'-D'^{2})z=\cos(3x+y)$ where $D=\partial /\partial x$ and $D'=\partial/\partial y$ appeared on my exam, and I'm trying to check whether the ...
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Fredholm integral equation of the second kind - no homogeneous cases anywhere?

I'm trying to solve this homogeneous Fredholm integral equation of the second kind for $y(x)$. $y(x) = \int^{\infty}_{-\infty} \ln \left( \frac{1}{x-t} \right) y(t) dt$ I'm aware of Liouville-Neumann ...
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Finding the basis vector of null space, how?

Context from this video and the lecturer did not proceed to solve for the basis vectors of this null space.: I need to find the basis vector of the null space and I get that we need to first find the ...
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How do free variables become a constant that represents all span of a vector?

Context: From this video(timestamped), making sense of null spaces: I understand: First statement from the left that we turned $x_2 = \begin{bmatrix}1 \\ 0\end{bmatrix} = s$ (yellow column). second ...
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How to solve homogeneous differential equation with initial value conditions using Green's function?

Solve the differential equation $$xy'' + y' = 0$$ using the Green’s function satisfying the initial condition $y(1) = y'(1)$. Generally, Green's functions are used to solve nonhomogeneous differential ...
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Solutions for the composition of homogeneous ODE

Suppose we have a second-order homogeneous ODE represented by $$ F(x,y,y',y'')=0 \quad (1), $$ and then consider the $4th$ order homogeneous ODE \begin{equation} F(t,F(x,y,y,y''),F(x,y,y',y'')',F(x,y,...
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Homogenous Initial Value Problem's

I've been tasked with the following. Consider $x^\prime(t)=(x^2(t).t^2)/(tx(t))$ with $x(1)=1$. a) Show the IVP is homogeneous. b)Find x: I $\longrightarrow$ $\mathbb{R}$ specifying the maximal ...
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Equilibrium point rumor propagation model

$$ \begin{aligned} I' &= -bkIS \\ S' &= bkIS - akS(S+R) \\ R' &= akS(S+R) \end{aligned} $$ where $I'=S'=R'=0$. The inspection of equation system indicates that equilibrium stated are only ...
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$x^2\frac{dy}{dx}+xy+1=0$

Solve the following differential equation by the form of homogeneous equation. Letting $y=vx$ The equation: $x^2\frac{dy}{dx}+xy+1=0$ I can’t separate variables My solution steps are: $x^2(v+x \frac{...
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Show that an equation with $log^2(\cdot)$ is homothetic?

I need to show that the following function $v$ is homothetic (i.e., that there exists a strictly increasing function $g: \mathbb{R} \rightarrow \mathbb{R}$ and a homogenous function $u: \mathbb{R}^n \...
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Integrating factor in a homogeneous equation

(From Boyce and Di Prima Book). Prove that if $$M(x,y)\,dx+N(x,y)\,dy=0$$ is an homogeneous equation, then one of its integrating factors is: $$ \mu(x,y)=\frac{1}{x\,M(x,y)+y\,N(x,y)}$$ Please help ...
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Transition in solving differential equations using Method of Undetermined Coefficient

Given the following equation: $x^2y''\:-\:3xy'\:+\:3y\:=\:lnx$ When I let $lnx$ be $z$, I am not able to understand the transition that yields: $y''\:-\:4y'\:+3y\:=\:z$ I tried using things I know: $z ...
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Retarded Solution in Duhamel's Principle

According to Wikipedia's explanation of Duhamel's principle, the retarded solution $P^s f$ is the particular solution to a homogeneous (but with inhomogeneous initial condition) version of an ...
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Can a non-homogenous augmented matrix be converted to homogenous and then be solved using Gaussian elimination?

Pretty much the title. I want to know if we can convert a non-homogenous linear equation into a homogenous one. I'm specifically asking about the equations that can be turned into an augmented matrix (...
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Proof for transforming a non-homogenous linear recurrent into a homogenous linear recurrence.

I need to prove that $b^np(n)$ becomes $(r-b)^{d+1}$ as shown in the theorem below. A nonhomogeneous linear recurrence of the form $$a_0t_n + a_1t_{n-1} + \cdots + a_kt_{n - k} = b^np(n)$$ can be ...
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Unable to solve the following question [closed]

I am trying to solve the following question but I can not figure out a method to solve it. The question is: $$x+\sqrt{(y^2) - (xy) \frac{dy}{dx}} = y, y(1/2) = 1$$ Can somebody help me and guide me ...
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Find the condition such that $f$ is homothetic

If $f=x^{\frac{1}{3}}y^a+x^{\frac{2}{3}}y^b$, $a,b>0$ and $x,y>0$, then what are the conditions under which $f$ is homothetic? I know how to find the conditions such that $f$ is homogeneous but ...
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Finding the null space of given linear transformation $T: \mathbb{R}^{4}\mapsto \mathbb{R}^{3}$.

In my Linear Algebra test, I was supposed to find the null space of the linear transformation $T:\mathbb{R}^{4}\mapsto \mathbb{R}^{3}$, which is defined as follows: $$T\begin{bmatrix}x_{1}\\ x_{2}\\ ...
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Piece-wise second order differential equation

Suppose I have some function $f(x)$ which is equal to $kx$ when $x$ is negative and $-kx$ when $x$ is positive, and I have the following differential equation : $$\frac{d^2x}{dt^2}=-k|x|=f(x)$$ My ...
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Using Undetermined Coefficients find the particular solution

Given that $xy^{\prime\prime}+(2+2x)y^{\prime}+2y=0$ has a solution $y_{1}(x)=x^{-1}$, find the general solution of the differential equation $xy^{\prime\prime}+(2+2x)y^{\prime}+2y=8e^{2x}$ I used ...
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Solve $y''' + 4y'' + 5y' = 0$

Find the solution to the $y''' + 4y'' + 5y' = 0$ using the wronskian. What I have done: $$y^3+4y^2+5y=0 \\ \implies y(y^2+4y+5) \\ \implies y^2(y+4)+5y=0 \\ \implies y+4 = -\frac{5}{y} \\ \implies \...
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Homogeneus nonlinear function in multiple dimensions

In this question, somebody asked about a homogeneous but nonlinear function. The answer in general form was: "Actually the most general function with the desired property would be $f(x)=\alpha x+\...
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How to prove the solutions given are fundamental set of solutions without a given differential equation?

We are tasked to prove the two solutions below are two fundamental set of solutions. These are the two solutions given to us and nothing else. $y_1(t) = t^{-1}$ $y_2(t) = t^{3/2}$ All I've learn so ...
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What is the degree of Homogeneity

I have the function $Q = L + K + L^{0.5}K^{0.8}$ I am struggling to find the degree of homogeneity, any ideas on the answer?
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What is the geometric interpretation of the normal equations $\mathbf{N}={{\mathbf{A}}^{T}}\mathbf{A}$ matrix when the columns of A are orthogonal?

I have a homogeneous problem with the form $\mathbf{A}_{n\times6}\mathbf{x}_{6\times1}=\mathbf{0}_{n\times1}$ in which the columns of the $\mathbf{A}$ are orthogonal in each row, for example $\mathbf{...
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Solution of the ordinary differential equation $y''+\sin(x)y'+2y=1$

I have the following ordinary differential equation before me: $y''+\sin(x)y'+2y=1$. I have to find its PI(particular integral).My strategy is to come up with the solution of corresponding homogeneous ...
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2 votes
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Power Series Solution of Non-homogeneous Differential Equation: $(1-x^2)y'' + y' + y = xe^x$

I am attempting to solve the non-homogeneous differential equation $(1-x^2)y'' + y' + y = xe^x$ via power series solution, but am running into an issue once I have expanded all the necessary terms and ...
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Solve the following recurrence $\left(n+1\right)a_n=na_{n-1}+1\:\:\:\:\:\forall n\ge 1\:and\:a_0\:=\:1$

I am trying to resolve this recurrence $\left(n+1\right)a_n=na_{n-1}+1\:\:\:\:\:\forall n\ge 1\:and\:a_0\:=\:1$ What i have tried is to First write the homogeneous equation $a_{nh}=\frac{n}{n+1}\cdot ...
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Doubt in a lemma in Hoffman Kunze from System of equations

There is a lemma in hoffman Kunze which states that: If A is a $m \times n $ matrix where $m<n$ then it has a non-trivial solution. I understand the proof of this which uses the fact that the ...
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2nd Order Differential Equation (Homogenous & Variable Coefficients)

I have the following differential equation: $y''(x) + \frac{1}{x}y'(x)+\frac{\lambda x}{D}y(x) = 0$ From what I understand, DE's of this kind have a general solution of: $y(x)=Ay_1(x)+By_2(x)$ but at ...
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Proving $\sin^2(x)+\cos^2(x)=1$ using Abel's Theorem / Deriving Pythagoras Theorem from Abel's Theorem

I've been trying to prove the following, $\sin^2(x) + \cos^2(x) = 1$ from Abel's theorem, and from the fact that $y_1 = \cos(x)$ and $y_2 = \sin(x)$ form a set of fundamental solutions for the ...
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Integration by parts to solve homogeneous variable coefficient differential equation?

Define $u(x',y')$ the solution of a differential equation. Solve $x'u_{x'}=0$ Do I Solve using integration by parts?
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Solving differential equation with products

Consider the equation $$\delta e^{\delta x}+ \frac{(1-e^{\delta x})(-ae^{-ax}+(a+\delta)e^{(a+\delta)x} +\delta e^{\delta x})}{(1+e^{-ax}+e^{(a+\delta)x}+e^{\delta x})} =0$$ Denoting $g(x)=1-e^{\delta ...
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Significance of homogeneous equations

I'm studying ODE. Over my math education career, I've occasionally encountered reference to homogeneous equations. Now, in ODE, I'm learning how to solve differential equations that are homogeneous. I'...
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How do I prove the following differential equation is homogenous of degree 0?

I understand that if I have a differential equation that I can write in the form of $$ A(x,y)dx+B(x,y)dy=0 $$ , where $A(x,y)$ and $B(x,y)$ are both homogenous functions of the same degree, then I ...
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converting to minutes

I'm trying to answer a question in minutes but I just can't seem to be able to eliminate all the other units. ((B×D)/(M×PWR))×60 B = 500Wh, D = 80% of B, M = 12.505kg, PWR = 55W/kg I should be able to ...
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Prove that a distribution $F$ is homogeneous of degree $k$ if and only if $\sum_{j=1}^dx_j \frac{\partial F}{\partial x_j}=kF$

Taking derivatives both sides with respect to $a$ in the definition of homogeneous distribution equation $$a^{-d}\int f(x)\phi(\frac{x}{a})=a^k\int f(x)\phi(x)$$ for all $a>0$ and the test function ...
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Reduction of Order in Homogeneous Linear ODE

I was trying to solve this exercise, $yy''=3(y')^2$ The question asks to reduce the ODE to first-order before proceeding to solve the equation. For your information, I am still in Homogeneous Linear ...
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