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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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What is the particular $Y_p$ for the general solution $y''+y=6\sin x$?

In this case, We set $Y_p=Cx(A\cos x+B\sin x)$ beacuse of $D= \pm i$ then, $$Y_p''=2C(-A\sin x+B\cos x)+Cx(-A\cos x-B\sin x)$$ $$Y_p''+Y_p=2C(-A\sin x+B\cos x)=6\sin x$$ So $AC=-3$, BC=0, if C=1, then ...
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particular solution for D²y+4Dy+4y=18coshx?

set $y_p=K\cosh x$. $y_p'=K\sinh x$, $y_p''=K\cosh x$ inserting those functions into the original equation, then... $K\cosh x+4K\sinh x+4K\cosh x=18\cosh x$ the coefficient of $\cosh x$ would be 5K=18,...
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Linear homogeneous second order ODE with a non-constant coefficient

I am having problem on solving ODE $y'' - \Big( \frac{b^2}{c^2} + \frac{b}{c^2}\cdot \mu(x)\Big)y = 0$, where $\mu(x) \geq 0$ for all $x$, $b$ and $c$ are constants and $c > 0$. I don't really ...
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Reference on higher-order coupled system of ODEs

I have the system of $m$ coupled ODEs given as: $$A_4 x^{(4)} + A_3 x^{(3)} + A_2 \ddot{x} + A_1\dot{x}+A_0x =0$$ along with $4m$ initial (i) and final (f) conditions $x_i$, $\dot{x}_i$, $x_f$, ...
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Proof of reduction of non-homogeneous ODE problem to homogeneous

Consider the following boundary value problem: $$-(p(x)u')'+q(x)u=g(x) \hspace{35pt} (1)\\ 0<x<1, u(0)=u_l, u(1) = u_r\hspace{35pt}$$ where $p\in C^1[0,1], p(x) \ge p_0 >0; q\in C[0,1], q(x)\...
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Define the set $E(\mathbb{Q})$ of $\mathbb{Q}$-rational points on an elliptic curve

I'm a bit struggling with defining the set of $\mathbb{Q}$-rational points on an elliptic curve $E:\;y^2=x^3+ax^2+bx+c$ with $a,b,c\in\mathbb{Q}$. I'm actually guessing that If we let $K$ and $L$ be ...
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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}, \...
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Minimum value of a general two degree curve

Let F(x, y) = $ax^2 + by^2 + 2gx + 2fy + 2hxy + c$ To find its minimum value I use the following logic abd procedure : When you put a point (x, y) in equation of a conic it gives a positive value when ...
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2nd Order ODE (Variation of Parameter)

I am given the equation $$ y''+y = x\cos x - \cos x $$ with initial values of $$y(0)=1, y'(0)= 1/4,$$and I believe this needs to be solved using the method of variation of parameters, though I'm ...
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Linear dependence with homogeneous 3rd order DE

I am trying to figure out if this equation is solvable. I was given $$ y^{(3)} + 3y^{''} - 10y^{'} = 0 $$ with initial conditions $$ y(0)=7, y'(0)=0, y''(0)=70. $$ I solved the characteristic ...
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Solving $a_2(t)\frac{d^2u}{dt^2}+a_1(t)\frac{du}{dt}+a_0(t)u=f$

Let $$Lu=f$$ be $$a_2(t)\frac{d^2u}{dt^2}+a_1(t)\frac{du}{dt}+a_0(t)u=f$$ Let $u_0$ be a solution for $Lu=0$. Assume that the solution $u$ is of the form $u=u_0v$ a) from the equation $Lu=...
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Finding the general solution for a higher order different equation with constant coefficients

I am given $y^{(8)}+y^{(6)}+y^{(4)}+y^{(2)}+ay=0$ where $a$ is a real number. The question asks for which values of $a$ does the equation have a real-valued solution which is never $0$. Convert to ...
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Zeroes of homogenous polynomial different of 0 at $\mathbb{R}^2$

We consider the following differentual equation, $$x'=P_n(x,y)$$ $$y'=Q_n(x,y)$$ Where $P_n$ and $Q_n$ ara homogenous polynomials with degree $n$. How do I proof that there's only one critic point and ...
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Finding the roots of a 2 variable polynomial

Consider the polynomial $x^{n+1}+y^{n+1}+yx^n$ where $n\in\mathbb{N}$ is odd. I would like to find the real roots of this polynomial. I believe the only root is $(x,y)=(0,0)$. So far, I've been able ...
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1answer
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Proving the existence of a center or a focus in a system of ODEs

I'm asked to prove that for any odd value of $n\in\mathbb{N}$ there exist homogeneous polynomials $P_n,Q_n\in\mathbb{R}[x,y]$ such that the point $C=(0,0)\in\mathbb{R}^2$ is either a center or a focus ...
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When solving differential equations using substitution, does it really matter which substitution you choose?

My test says to make the "appropriate substitution," but one of the differential equations can be solved as a homogenous substitution and a bernoulli's substitution. Both yield different results. Is ...
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Homogeneous 1st Order Differential Equations

I have come across two different definitions of a homogeneous 1st order ODE; that the equation can be written in the form $y' = f(\frac{y}{x})$, and that in the form $M(x, y)dx + N(x, y)dy = 0$, it is ...
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solving a certain homogeneous ODE

I want to solve the homogeneous first-order ODE $$y'=\sin(y/x)+(y/x)$$ Using substitution $v=y/x$ we separate variables to get $$\csc(v)\;dv=1/x\;dx$$ and hence $$|\csc(v)+\cot(v)|^{-1}=A|x|,\;\;\;A&...
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Visualising dot product P.l in Projective geometry

I am trying to understand how the following identity in homogeneous coordinates comes about and how to visualise the two multiplying parts P and l in the 3d cartesian coordinates. Are they orthogonal ...
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1answer
54 views

Solving PDE using separation of variables (Heat diffusion)

I am trying to solve a standard PDE, but I got stuck on how to choose the separation constant such that I do not end up with a trivial, uninteresting solution. The system is presented below and my ...
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1answer
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Finding homogennous and particular solution of an equation

Let's say I have an equation like this: $Sn = 2 \cdot S(n-1) + 2^{(n+1)} - 2$ Now I know that I need to transform it to look like this: $S(n+1) - 2 \cdot Sn = 2^{(n+2)} - 2$ From this I can get ...
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Partial Differential Equations Question: State if the following PDEs are linear homogeneous, linear nonhomogeneous, or nonlinear:

State if the following PDEs are linear homogeneous, linear nonhomogeneous, or nonlinear: $u_{t}+u_{x}=sin(x)u$ $u_{tt}-u_{xx}=e^{t}u_{t}$ $u_{tt}-u_{xx}=x^{2}$ $u_{xx}+u_{yy}=u_{x}u_{y}$ As for ...
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Solve the recurrence relation $a_n=a_{n-1}+2n+3a_{n-3}?$

I am currently solving a recurrence relation but I got stuck since I am not even able to find the basic root of the auxiliary equations.
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1answer
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Solving the Airy Equation using Laplace Transform

I have been trying to read the book Airy Functions and Applications to Physics, Olivier Vallee & Manuel Soares, for research on the Airy Functions, but I am stuck on using the Laplace transform to ...
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1answer
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For row reduced echelon matrix for a homogeneous system of equations, what solution would be there for r=n and r>n

I was learning Linear Algebra from Hoffman and Kunze. There the authors prove that for a row reduced echelon matrix with rows r and columns n for a homogeneous system of equations X, if r < n, X ...
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Riccati and Linear 1st Order ODE Parallel

I have noticed a certain similarity between Riccati 1st Order ODEs and linear 1st Order ODEs. Specifically, the general solution for each is given by any particular solution plus some function of the ...
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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: $$\frac{dy}{dx}=\frac{1}{y^3}\frac{\left(1+2y^4\right)}{1+\frac{4x}{y^4}}$$ ...
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1answer
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Homogeneous definition for first order differential equation and higher order differential equations?

For first order https://en.wikipedia.org/wiki/Homogeneous_differential_equation#Homogeneous_first-order_differential_equations and for higher order, it is https://en.wikipedia.org/wiki/...
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Linear polynomial whose multiplication with a given quadratic polynomial vanishes in the Jacobi ring

Let $\mathbb C[x_1,\ldots,x_n]$ be the polynomial ring of $n$ varibles, and $\mathbb C[x_1,\ldots,x_n]_d$ be its homogeneous degree $d$ piece. Let $F\in \mathbb C[x_1,\ldots,x_n]_3$ be a homogeneous ...
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Having a homogenous system of linear equations with real coefficients with a non-trivial complex solution than there is a real solution too.

I'm struggeling a bit with this proof. Suppose we have a homogenous system of linear equations with real coefficients with a non-trivial complex solution than there is a real solution too. This ...
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Homogeneous difference equation

I have to solve two different homogeneous difference equations. 1) $z_0 = 2 , z_1=1 , z_{j+2} - z_{j+1} -z_j = 0 $ and 2) $z_0 =2 , z_1=0 , z_2 =16 , z_3=-14 ,$ $z_{j+4} - 6z_{j+2} +8z_{j+1} - ...
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3answers
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Find particular solution of $u''$ + 2$u'$ + $u$ = 9$x^2$ $e^{-x}$.

Find particular solution of $u''$ + 2$u'$ + $u$ = 9$x^2$$e^{-x}$. So I've already worked out that the particular solution is A$e^{-x}$ + B$xe^{-x}$ so then I tried the particular solution $u_p$ = C$x^...
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1answer
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Boundary conditions in linear homogeneous differential equation with constant coefficients

I have the linear homogeneous recurrence relation: $p_{k+1} -2 p_k + p_{k-1}=0$, where $p_k$ - probability of getting $N=5$, starting with $k=3$. I'm trying to find boundary conditions, which are ...
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1answer
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finding the general solution to inhomogeneous ODE

$$u'' + 4u= 8x^2 + 13e^{3x} + 16\cos2x.$$ So to find the general solution I first have to find the homogeneous and particular solutions and then general solution will be $u= u_h + u_p$. So ...
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How to prove these differential equations from the homogeneity relation?

Let $N(x,\bar x)$ a real function of $x=(x_1,\cdots,x_n)\in \mathbb{C}^n$ and its conjugate $\bar x=(\bar x_1,\cdots,\bar x_n)\in \mathbb{C}^n$ that satisfies the homogeneity relation $N(\lambda x,\...
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Is this problem wrong? I keep getting $y = ±(\sqrt{c-x^4})/x$ which is not possible as $x = 0$ and $y = 2$. Am I wrong?

Find the particular solution that satisfies the initial condition: Differential Equation: $$(2x^2+y^2)dx+xydy=0$$ Initial Condition: $$y(0) = 2$$ I solved it and got $$y = ±(\sqrt{c-x^4})/x$$ ...
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Can following Differential equation be reduced to Homogeneous Differential equation?

Question: ($x^2+2xy^2)dx + (2x^2y + y^2)dy = 0$ or, $\frac{dy}{dx}$ = $\frac{-(x^2 + 2xy^2)}{(2x^2y + y^2)}$ Here, the given Differential equation(D.E.) is not Homogeneous. Since,this question has ...
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Dimensionally Homogenous question

"Consider the mass of algae in the lake. The mass M of algae at some time moment t is described by the following equation: $dM/dt$ = kM −λ with the initial condition M(0) = M0, (B) where k is the ...
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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} (*)...
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1answer
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How to get parameter value given the form of the homogeneous differential equation solution

The question is: if I know the differential equation has a solution, which has the form of the quadratic polynomial, how do I get to solve the unknown of the equation? For example, for the ...
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System of homogeneous equations and irreducibility

Is a system of homogeneous polynomial equations of n eqs and n+1 variables, over complexes, always irreducible? Any reference?
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Knowing which homogeneous equation to assume with nonhomogeneous differential equations

When it comes to the method of undetermined coefficients, I am having a difficult time figuring out which homogeneous equation to assume when it comes to nonhomogeneous differential equations. For ...
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How to solve this equation (maybe Helmholtz equation)?

How can I solve this equation? I think this is an inhomogeneous Helmholtz equation, however, the sign of $m^2$ is negative. $\nabla^2\delta R(\vec{r})-m^2\delta R(\vec{r})=\frac{1}{3f_{RR}}\bigg(\...
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2answers
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Free,Undamped Mechanical Vibrations

In the case of free, undamped vibrations,the differential equation is $mu''+ku=0$ and solution to this differential equation is \begin{align} \tag{1} u(t)=c_1\cos{(\omega_0 t)}+c_2\sin{(\omega_0 t)} \...
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1answer
71 views

Solve the Differential equation $\frac{dy}{dx}=\frac{5x^3-xy^2-2x}{3x^2y-y^3}$

Solve the Differential equation $$\frac{dy}{dx}=\frac{5x^3-xy^2-2x}{3x^2y-y^3}$$ My try: Let $x^2=X$ and $y^2=Y$ we get $xdx=dX$ and $ydy=dY$ then $$\frac{dY}{dX}=\frac{5X-Y-2}{3X-Y}$$ which is a ...
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Residual error of a normalized DLT

I use the Direct Linear Transform (DTL) to estimate a homography from a given set of point correspondences $(x_i,x_i')$, such that $$x_i' = \mathbf{H}x_i$$ Therefore, I set up the measurement matrix $\...
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3answers
207 views

Solving $y''+2y'+2y=e^{-x}\sin(x)$ using variation of parameters.

I am stuck on the following differential equation. Here is where I got to. We first solve the homogeneous equation. \begin{align*} &y^{\prime\prime} + 2y^{\prime} + 2y = 0\Longrightarrow r^2+2r+2=...
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2answers
38 views

How to solve this Homogeneous differential equation? [closed]

$$ \frac{4xy}{(x^2-y^2)}\frac{dy}{dx} = 1$$ when $y=0 , x=1 $ show that $$ \sqrt{x}.(x^2-5y^2) =1 $$ using this substitute $ y=vx $.
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Solution to this 2nd order linear homogeneous ODE

How do I find a solution to: $$i p(t)\ddot x-\left[p(t)^2+i\dot p(t)\right]\dot x+2ip(t)^3x=0\ ,$$ where $p(t)$ is some chosen smooth function, and the overdot represents derivative with respect to $t$...
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0answers
19 views

Show that if a real homogeneous system of two-first order equations…

Showing that if a real homogenous system of two-first order equations has a fundamental matrix, then there is another matrix that is also a fundamental matrix So, given a fundamental matrix for a ...