Homogeneous equation Definition and 34 Threads

Ignoring the second part of the question for now, since I think it will be more clear once I understand how this equation is homogeneous. According to my textbook and online resources a first-order ODE is homogeneous when it can be written like so: $$M(x,y) dx + N(x,y) dy = 0$$ and ##M(x,y)##...

I'm having quite a bit of a problem with this one. I've managed to figure out that ##T_0 = 0##. However, not knowing what ##q(t)## is bothers me, although it seems that I could theoretically solve the problem without knowing it. For ##t>t_1##, integration by parts gives me ##T = Ce^{-t/10}##...

Hi, I'm going to cite a book that I'am reading Can anyone provide some simple references where I can find at least an intuition regarding what is stated by the author. Thanks, Ric

Hi, I have solved this ODE till half way and got stucked on the integration of some weird expression. Need help for this. Thank you!

Hi, I have attached part of my steps for solving the homogeneous equation. The equation is proven to be homogeneous. However after using substitution of y=zx and its' derivative, I was not able to separate the variables conveniently as shown. Please advise. Thank you!

For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?

Hello, If I have a homgeneous linear differential equation like this one (or any other eq): $$y''(x)-y'(x)=0$$ And they give me these Dirichlet boundary conditions: $$y(0)=y(1)=0$$ Can I transform them into a mixed boundary conditions?: $$y(0)=y'(1)=0$$ I tried solving the equation, derivating...

Homework Statement Find the general solution of y^{(5)}-y(1)=x The Attempt at a Solution I found the complementary function by substitution of the solution form y=e^{kx} giving k=0,1,-1,i,-i, so y_{cf}=a_0+a_1e^x+a_2e^{-x}+a_3e^{ix}+a_4e^{-ix} Now for the particular integral, the general...

Homework Statement Show that the homogeneous equation: $$(Ax^2+By^2)dx+(Cxy+Dy^2)dy=0$$ is exact iff 2b=c. Homework Equations None, just definitions. The Attempt at a Solution Let $$M = Ax^2+By^2$$ and $$N = Cxy+Dy^2$$ Taking the partial derivative of M with respect to y and the partial of...

Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...

Suppose ##x_1(t)## and ##x_2(t)## are two linearly independent solutions of the equations: ##x'_1(t) = 3x_1(t) + 2x_2(t)## and ##x'_2(t) = x_1(t) + 2x_2(t)## where ##x'_1(t)\text{ and }x'_2(t)## denote the first derivative of functions ##x_1(t)## and ##x_2(t)## respectively with respect to...

Hello. I forgot the reason why 2nd order differential equation has two independent solutions. (Here, source term is zero) Why 3 or 4 independent solutions are not possible? Please give me clear answer.

Homework Statement Solve: A*sin(ωt + Θ) = L*i''(t) + R*i'(t) + (1/C)*i(t). Where: A=2, L = 1, R=4, 1/C = 3 and Θ=45°. Homework Equations The system has to be solved by i(t) = ih + ip. I gave the values to A, L, R, 1/C and Θ. I can also give values to ω, but I've come to a doubt when solving...

Homework Statement The system is declared as follows: 8/(2*x - y) - 7/(x + 2*y) = 1 4/((2*x - y)^2) - 7/((x + 2*y)^2) = 3/28 Homework Equations The Attempt at a Solution I define 'x' to equal k*y and I replace it inside the equation: 8/(2*k*y^2) - 7(k*y + 2*y) = 1...

I need help finding a linear homogenous constant-coefficient differential equation with the given general solution. y(x)=C1e^x+(C2+C3x+C4x^2)e-x 2. I tried to come with differential equation but this is it I can 't seem how to begin

Actually I can't find if a differential equation is homogeneous or not I thought homogeneous is given by dy/dx= f(x,y)/ g(x,y) but it doesn't look like that For eg: dy/dx= (y+x-1)/(y-x+2) is not homogeneous at all though f(x,y)=y+x-1 and g(x,y)=y-x+2 How can you tell...

My doubt is that is dimension of a 2nd order homogeneous equation of form y''+p(x)y'+q(x)=0 always 2 ? or dimension is 2 only when p(x),q(x) are contionuos on a given interval I..??

Hi, I don't understand why the general solution of 2nd order homogeneous equation is linear? Why is c_1e^(xt)+c_2e^(xt) a linear differential equation? What am I missing here? Any help would be appreciated, I'm struggling a bit understanding the concepts of differential equations...

Homework Statement xdx+sin\frac{y}{x}(ydx-xdy) = 0 The Attempt at a Solution Well, it's quite easy. But I'm quite confused if this is homogeneous or not, because of the sine function. This is my solution, assuming that this is a homogeneous equation. let x = vy, dx = vdy + ydv; then...

Homework Statement The problem is setting up the equation, it says that the matrix equation will be made up of four equations for the 2 unknowns. I'm supposed to find for which a's and b's the equation is true, using a linear system and gaussian elimination. Homework Equations A2 + aA + bI2 =...

i am having trouble finding the general solution for the given homogeneous equation: x2yy' = (2y2 - x2) which i made into x2dy = (2y2 - x2) dx i turned it into the following: (2y2 - x2) dx - x2 dy = 0 then i used substitution of y = xv and got (2(xv)2 - x2 - x2v) dx - x3 dv = 0 then...

Homework Statement The problem has to do with diagonalizing a square matrix, but the part I'm stuck on is this: Bx=0, where B is the matrix with rows [000], [0,-4, 0], and [-3, 0, -4]. After performing rref on the augmented matrix Bl0, I get rows [1,0,4/3,0], [0,1,0,0], [0000]. I am...

Homework Statement y'' + y = -2 Sinx Homework Equations The Attempt at a Solution finding the homogeneous solution, is simple; yh(x) = C1 Cos(x) + C2 Sin(x) for the particular solution, I let y = A Cos(x) + B Sin(x) thus, y' = -A Sin(x) + B Cos(x) y'' = -A Cos(x) - B...

Homework Statement Find y as a function of x if y'''−11y''+28y'=0 y(0)=1 y'(0)=7 y''(0)=2 I have one attempt left on this question. Could someone verify my answer for me? Homework Equations The Attempt at a Solution (use t as lamda) t^3-11t^2+28t=0...

Homework Statement (4y4-9x2y2-144)dx - (5xy3)dy = 0 Homework Equations substitute y = xv dy = dx v + dv x The Attempt at a Solution after substituting i got (4x4v4-9v2x4-14x4)dx - (5v3x4)dx.v + dv.x = (4v4-9v2-14)dx - 5v3(dx.v + dv.x) = 0 = dx(4v4-9v2-14-5v4)+dv(-5v3x)= 0...

how does one use the determinant of the coefficient matrix of a system to determine if the system has nontrivial solutions or not?

Homework Statement y2dx -x(2x+3y)dy =0 I have to recognize the equation and solve it Homework Equations The Attempt at a Solution I did y2dx - (2x2+ 3yx) dy=0 which is a homogeneous now after I substitude x=uy dx=udy + ydu I stuck here after the substitution...

First Order Linear Non-Homogeneous Equation Homework Statement I need to solve for e(t) Homework Equations Do I use Laplace Transform for the last integral? The Attempt at a Solution \begin{subequations} \begin{eqnarray} \nonumber \dot{\hat{{\cal E}}}(t) &=& -\kappa...

this is the given equation y'=(4y-3x)/(2x-y) and here is all the work I've done so far: (4v-3)/(2-v)=v+x*dv/dx i moved v over and came up with this (-3+2v+v^2)/(2-v)=x*dv/dx did a flip (2-v)/(-3+2v+v^2)dv=dx/x by partial fractions I got a=-3/2 and b=1/2 so...

Homework Statement Given, (y+2)dx + y(x+4)dy = 0, y(-3) = -1Homework Equations v=y/xThe Attempt at a Solution I've been REALLY struggling with homogeneous equations for some reason...I just don't understand them all. so far I've tried two things. (1)dx -(y)dy ----- ------- (x+4)...

y''(t)+A^2y(t)=f(t), t>0, y(0)=B, y'(0)=C, A, B, C\in\mathbb{R} e^{iAt} is a particular solution of the homogeneous equation. I can multiply it by some arbitrary function and find another solution of the homogeneous case, but when I try with the f(t) on the RHS, I can't do it. Anyone help?

Hey... So the question is as stated: Show that \frac{1} {M_x + N_y} , where M_x+N_y is not identically zero, is an integrating factor of the homogeneous equation M(x, y)dx+N(x, y)dy=0 of degree n. So I am not too sure where to go with this. I suppose what it's saying is, that I'm...

(x^2 + y^2)dx + (2xy)dy = 0 I get y = sqrt((kx^5 + x^2)/3) Where k = c2 cubed, and c2 = ln(c) so k = 3ln(c) But, the answer the teacher gave is (x^2)(y^3) - x - ln(y) = c I can't come up with anything remotely close. I know this isn't in a pretty LaTeX form, but I am new and haven't...

Can someone explain what the homogeneous equation is :redface: and how do you find the 'null vectors' and hence the general solution. Eg. AX = [6] [8] [4] A = [1 2 4] [3 1 2] [0 2 4] X = [2] [0] [1] Find the null vectors of A and general solution.