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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Finding a system of homogeneous linear equations given solution space

If, in $R^4$, $\alpha_1 = (-1,0,1,2)$, $\alpha_2 = (3,4,-2,5)$, $\alpha_1 = (1,4,0,9)$, is given, How can I find a system of homogeneous linear equations with solutions space which is exactly spanned by above three vectors? I don't know even how to…
smw1991
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A question about "General and Particular Solutions" (I guess) which I'm not sure what they mean

We have a $3\times 6$ matrix $A$ with rank $3$ (this is all the information we have, no matrix given). Here comes the questions: What is the number of free variables in the solution to the system $Ax = 0$? (Well, it must be 3. Although I think I…
Kerem
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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 series expansion of any function or not. I know the…
Shivam Kumar
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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 transformed into a homogeneous linear recurrence…
SkiCode87
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Homogeneous linear DE general solution

I need help with a homework question. I am clueless about this. Too many roots. I don't know what to do. Please help me with the complete solution. Please ignore the definition part. Picture link
Md Mosleh Uddin
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How can I show differential equation of the form can be transformed into a separable differential equation?

Here is my question: If $H(x,y)$ is a homogeneous function of degree n, show that the differential equation of the form $$y^nf(x) + H(x,y)(ydx - xdy) = 0$$ can be transformed into a separable differential equation. How can I solve it?
colss
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How to write the General solution of the homogeneous system with complex entries in the Eigan vectors in terms of real functions?

I have a function: So I know that $ e^{it} = cos(t) + isin(t) $ I'm solving a homogeneous system and it has complex entries in 2 of the eigan vectors. So I need to re-write these in terms of real functions. So far I have this: $ y =…
Andrew Kor
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For which values is this function homogeneous?

Given the function $u=z^\alpha_1z^\beta_2+\gamma z_3+\delta$, for which non-negative values of $\alpha, \beta, \gamma, \delta$ is the function homogeneous? I am only finding an answer where $\gamma = 0, \delta = 0$ with degree $\alpha+\beta$, but…
William Fluck
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Homogeneous differential equation with undetermined coefficient

Given the problem: $y’’+16y=\cos(4x)$ It’s particular $y$ is equal to zero. I know how to get the complementary $y$ but I had problem with the particular $y.…
Bido262
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Solving Non-Homogeneous Equation (Not involving Differentials)

I don't understand why the following non-homogeneous equation has been solved as shown below: $h(x) = 1 + ph(x-1) + (1-p)h(x+1)$ Let $h(x) = Rx$ Hence, $Rx = 1 + pR(x-1) + (1-r)R(x+1)$ Which gives, $R = \frac{1}{2p - 1}$ Hence $h = \frac{x}{2p -…
user131983
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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 $f_{xx}(x, y) f_{yy}(x, y) = (f_{xy})^2$ Attempted solution I tried using…
user1255490
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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 $g(x)=1-e^{\delta x}$ and…
Spätzle
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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 cancel out Watts as one is a numerator and the…
Paul
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Missing step to my conversion of units

The problem I am trying to solve is converting 100m^3 to ft^3 100(m^3) (3.28ft/1m)^3 = 3529ft^3 What my brain is having trouble comprehending is (3.28ft/1m)^3... if I were to multiply only the units in the bracket it would be 3.28ft^3/1m^3 which…
Paul
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How can I see if this function is homogeneous

Say I have the function $$\tan(\frac{x}{y}) + x\log_{10} (\frac{y}{x}) \cos (\frac{x}{y})$$ I want to know if it is homogeneous. I know it has to satisfy $f(ax,ay)=a^kf(x,y)$, but I don't know if it does. A short answer/explanation would be…
user808757
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