# 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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### 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 ...
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 \... 0 votes 1 answer 22 views ### 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+\... • 123 0 votes 0 answers 11 views ### 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 ... 0 votes 0 answers 23 views ### 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? 1 vote 0 answers 39 views ### 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{... • 51 3 votes 1 answer 86 views ### 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 ... 2 votes 1 answer 122 views ### 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 ... 2 votes 6 answers 106 views ### 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 ... 3 votes 0 answers 37 views ### 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 ... • 1,408 0 votes 1 answer 50 views ### 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 ... 0 votes 0 answers 103 views ### 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 ... 0 votes 0 answers 28 views ### 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? • 383 0 votes 0 answers 24 views ### 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$$Denoting g(x)=1-e^{\delta ... • 131 1 vote 1 answer 98 views ### Significance of homogeneous equations I'm studying ODE. Over my math education career, I've occasionally encountered reference to homogeneous equations. Now, in ODE, I'm learning how to solve differential equations that are homogeneous. I'... • 131 2 votes 0 answers 45 views ### How do I prove the following differential equation is homogenous of degree 0? I understand that if I have a differential equation that I can write in the form of$$ A(x,y)dx+B(x,y)dy=0 $$, where A(x,y) and B(x,y) are both homogenous functions of the same degree, then I ... 0 votes 1 answer 26 views ### converting to minutes I'm trying to answer a question in minutes but I just can't seem to be able to eliminate all the other units. ((B×D)/(M×PWR))×60 B = 500Wh, D = 80% of B, M = 12.505kg, PWR = 55W/kg I should be able to ... • 23 0 votes 1 answer 84 views ### Prove that a distribution F is homogeneous of degree k if and only if \sum_{j=1}^dx_j \frac{\partial F}{\partial x_j}=kF Taking derivatives both sides with respect to a in the definition of homogeneous distribution equation$$a^{-d}\int f(x)\phi(\frac{x}{a})=a^k\int f(x)\phi(x) for all $a>0$ and the test function ...
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 ...