Ode Definition and 1000 Threads

Homework Statement Write an implicit Euler code to solve the system ##c'(x) = \epsilon c''(x)-kc(x)## subject to ##1-c(0)+\epsilon c'(0) = 0## and ##c'(1)=0##. Homework Equations Nothing out of the ordinary comes to mind. The Attempt at a Solution In the following code, there is central...

Homework Statement I want to show that $$f''(x) = g(x)$$ has a solution of the form $$f(x) = 2\int_0^{x} dx' (x-x') g(x').$$ It's not hard to verify that it is a solution, the question is how to find it. This should be easy and is likely a standard problem but I haven't found the right...

Homework Statement Find through ##O(\epsilon)##, for ##\epsilon \ll 1##, the solution to the system $$\epsilon x'(t) = -x+y\\ \epsilon y'(t) = -(\epsilon+1)y+x$$ Homework Equations ##x = \sum x_n\epsilon^n## and ##y = \sum y_n\epsilon^n## The Attempt at a Solution Substituting the...

Mod note: Member warned that the homework template is NOT optional find yp (particular integral) (D2 + 4D + 5) y = 2 e-2xcos(x) ((D+2)(D+2)+1) y = 2 e-2xcos(x) yp = [1/((D+2)2 + 1)] ⋅ 2e-2xcos(x) yp = e(-3)x ∫ ∫...

Homework Statement Solve to order ##\epsilon## $$\epsilon d_x(xd_xf)-xf=0$$ subject to ##|f(0)|<\infty## and ##f(1)=1## via matched asymptotic expansions. Homework Equations Nothing comes to mind. The Attempt at a Solution Perform a matched asymptotic analysis. In this case when I take a...

Homework Statement ## y''+\lambda y = 0 ; y(0) = 0, y(\pi)-y'(\pi) = 0## Homework EquationsThe Attempt at a Solution So, we have to test when lambda is equal to, less than and greater than 0. Let ## \lambda = 0## thus, the ODE becomes ## y'' = 0 ## which implies solutions of the form ## y(t) =...

Homework Statement Find the eigenvalues and eigenfunctions of the following boundary-value problem. ## y''+\lambda y = 0 ; y(0) = 0, y'(L) = 0 ## Homework EquationsThe Attempt at a Solution So, we have to test when lambda is equal to, less than and greater than 0. Let ## \lambda = 0## thus...

Homework Statement Given $$u''(x)+\lambda u = 0\\ u(-1)=u(1)=0.$$ If ##\lambda_0## is the lowest eigenvalue, show that ##4 \lambda_0 = \pi^2##. Homework Equations $$\lambda_0 = glb\frac{(L(u),u)}{(u,u)}$$ where ##glb## denotes greatest lower bound and ##L## is the Sturm-Louiville operator. I...

Homework Statement Write the following second-order ODE as a system of two first-order ODEs. ##d^2y/dt^2 + 5(dy/dt)^2 - 6y + e^{sin(t)} = 0## Homework Equations w = dy/dt The Attempt at a Solution The solution of the book says ##dy/dt = w, dw/dt = -5w - 6y + e^{sin(t)}##, but shouldn't it be...

Homework Statement When a rocket launches, it burns fuel at a constant rate of (kg/s) as it accelerates, maintaining a constant thrust of T. The weight of the rocket, including fuel is 1200 kg (including 900 kg of fuel). So, the mass of the rocket changes as it accelerates: m(t) = 1200 - m_ft...

Homework Statement ##\dfrac{dy}{dx} + y = f(x)## ##f(x) = \begin{cases} 2 \qquad x \in [0, 1) \\ 0 \qquad x \ge 1 \end{cases}## ##y(0) = 0## Homework EquationsThe Attempt at a Solution Integrating factor is ##e^x## ##e^x\dfrac{dy}{dx} + e^x y = e^x f(x)## ##\displaystyle ye^x = \int e^x...

Homework Statement ##(2x + 3y + 1)dx + (4x + 6y + 1) dy = 0## ##y(-2) = 2## Homework EquationsThe Attempt at a Solution Let ##z = 2x + 3y## then ##z^\prime = 2 + 3y^prime## ##\displaystyle \dfrac{(z + 1)}{2z + 1} + \dfrac13\left({dz \over dx} - 2\right) = 0## ##\dfrac{dz}{dx} = \dfrac{z-...

Homework Statement Given a set of fundamental solutions {ex*sinx*cosx, ex*cos(2x)} Homework Equations y''+p(x)y'+q(x)=0 det W(y1,y2) =Ce-∫p(x)dx The Attempt at a Solution I took the determinant of the matrix to get e2x[cos(2x)cosxsinx-2sin(2x)sinxcosx-cos(2x)sinxcosx-...

Hello, I just came across this forum and thought of asking for advice! I am reapplying to Engineering and I am taking summer school (Full term) and need to take 2 more classes to boost my avg. Right now, I am enrolled in Physics II, Calculus III, and ODE and I have to choose two out of the...

$\tiny{17.2.02}$ \nmh{1000} $\textrm{Solve the equation by the method of undetermined coefficients.}\\$ \begin{align*}\displaystyle y''-4y'&=\sin{x}\\ y_p&=A\sin{x}\\ \end{align*} $\textit{ answer}$ \begin{align*}\displaystyle y{\left (x \right )}& = C_{1} + C_{2} e^{4 x} - \frac{1}{17}...

Homework Statement The harmonic oscillator's equation of motion is: x'' + 2βx' + ω02x = f with the forcing of the form f(t) = f0sin(ωt)The Attempt at a Solution So I got: X1 = x X1' = x' = X2 X2 = x' X2' = x'' ∴ X2' = -2βX2 - ω02X1 + sin(ωt) The function f(t) is making me doubt this answer...

Homework Statement How many functions y(t) satisfy both y''+t^2*y=0 and y(0)=6? 2. The attempt at a solution As this is a second order differential equation, two initial conditions (for y and y') would be needed to obtain a unique solution (cf. existence and uniqueness theorem). So the...

Homework Statement We assume from ODE theory that given a smooth A: I → gl(n;R) there exists a unique smooth solution F : I → gl(n;R), defined on the same interval I on which A is defined, of the initial value problem F' = FA and F(t0) = F0 ∈ gl(n;R) given.(i) Show that two solutions Fi : I →...

Homework Statement Please bear with the length of this post, I'm taking it one step at a time starting with i) Let A: I → gl(n, R) be a smooth function where I ⊂ R is an interval and gl(n, R) denotes the vector space of all n × n matrices. (i) If F : I → gl(n, R) satisfies the matrix ODE F'...

Homework Statement The equation given: dy/dt = 3*y A basis for the space of solutions is required.The Attempt at a Solution According to me it is e^(3*t) but it has turned out false. Why? I am considering the answer "The basis is the set of all functions of the form c*e^(3*t) but a...

Hi there, I am modelling a four degree of freedom system which is the dynamics of two spur gears in mesh, having two rotational and two translation degrees of freedom, respectively, a diagram exhibiting the system can be seen below. I have derived the equations of motion (EOM) and...

Homework Statement Edit* should say F'(0) = F(0) = 0 Homework Equations I know that I typically need 3 equations for a 3rd order ODE, does this apply if the is no F'? In the picture above are the equations I came up with, am I on the right trail? Lastly I am familiar with RK4, however I have...

Homework Statement Let f : I → C be a smooth complex valued function and t0 ∈ I fixed. (i) Show that the initial value problem z'(t) = f(t)z(t) z(t0) = z0 ∈ C has the unique solution z(t) = z0exp(∫f(s)ds) (where the integral runs from t0 to t. Hint : for uniqueness let w(t) be another...

Homework Statement Find the solution of the differential equation by using appropriate method: t^{2}y^{\prime} + 2ty - y^{3} = 0 Homework Equations I'm thinking substitution method of a Bernoulli equation: v = y^{1-n} The Attempt at a Solution [/B] t^{2}y^{\prime} + 2ty - y^{3} = 0...

Hi, The problem is to solve: dy/dx = −[2x + ln(y)]*(y/x) Attempt: I have tried to see if it is exact, I found it not to be, I can't easily find a function to multiply by to make it exact either (unless I am missing something obvious). It clearly isn't seperable, nor is it homogenous (I know...

x2 + y + y2dx - x dy = 0 Integrating factor, I(x,y) = -1 / (x2 + y2) How to find the integrating factor ?Why I cannot use below method to solve the ode ? (1/N)(My - Nx) = g(x) , I(x,y)=exp( ∫ g(x) dx) OR (1/M)(My - Nx) = h(y) , I(x,y)=exp( -∫ h(y) dy)

Can someone check my work here? Both ##f=f(x)## and ##y=y(x)##. $$f'y'+\frac{fy''}{1+y'^2}=0\implies\\ \frac{y''}{y'(1+y'^2)}=-\frac{f'}{f}\\ \frac{y''}{y'(1+y'^2)}=-\ln(f)$$ Now let ##v=y'##, which implies $$...

Homework Statement \frac{dy}{dx}\:+\:ycosx\:=\:5cosx I get two solutions for y however only one of them is correct according to my online homework (see attempt at solution) Homework Equations y(0) = 7 is initial condition The Attempt at a Solution \int \:\frac{1}{5-y}dy\:=\:\int...

http://imgur.com/a/8QjoW http://imgur.com/a/8QjoW Hello- I am trying to determine the dynamics of this linearly-damped hinge. Assuming that: v(0) = 0 damping constant = b door has mass = m I was able to determine that: ∑Fx = -Fd * cos(45-θ/2) + Rx = m*dvx/dt ΣFy = -Fd * sin(45-θ/2) - Fg +...

Hi I am supposed to find solution of $$xy''+y'+xy=0$$ but i am left with reversing this equation. i am studying solution of a differential equation by series now and I cannot reverse a series in the form of: $$ J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ...$$ $$...

I am given the equations of Lorenz with respect to deterministic non-periodic flow: \frac{dX}{dt} = Pr(Y-X), X(0)=X_{0} \frac{dY}{dt} = -XZ + rX - Y, Y(0) = Y_{0} \frac{dZ}{dt} = XY-bZ, Z(0) = Z_{0} where Pr is the Prandtl number, r = Ra/Rac is the ratio of the Rayleigh number to its...

Homework Statement Consider the following problems In #2, they start the solution by saying: r(t)=u(t-1) in #3, they start by saying that r(t)=t-tu(t-1) I understand how to solve the problem once you get r(t), I just don't understand how they decide what r(t) is going to be.

Homework Statement I need to solve: x^2y''-4xy'+6y=x^3, x>0, y(1)=3, y'(1)=9 Homework EquationsThe Attempt at a Solution I know that the answer is: y=x^2+2x^3+x^3lnx Where did I go wrong. I was wondering if it's even logical to solve it as an Euler Cauchy and then use variation of parameters...

Homework Statement Consider the initial value problem x" + x′ t+ 3x = t; x(0) = 1, x′(0) = 2 Convert this problem to a system of two first order equations and determine approximate values of the solution at t=0.5 and t=1.0 using the 4th Order Runge-Kutta Method with h=0.1. Homework Equations...

Homework Statement How exactly they combined equation1 and equation2 and got that system? I don't get that part. Homework Equations A*(dy/dt)= -k*y eq1 A*(dz/dt)=ky-kz eq2 The Attempt at a Solution I tried substituting the 1st ky in the 2nd equation and then differentiating but I don't...

Homework Statement This is an ordinary differential equation using the differential operator. Given the system: d^2x/dt - x + d^2y/dt^2 + y = 0 and dx/dt + 2x + dy/dt + 2y = 0 find x and y equation Answer: x = 5ce^(-2t) y = -3ce^(-2t) Homework EquationsThe Attempt at a Solution We change...

Hi there. I have to solve a system of coupled ordinary differential equations. I have some initial values, but in different points of the domain. The equations are all first order. Let's suppose the system looks like this: ##\displaystyle\frac{dy_1}{dz}=y_1+y_2+0.01##...

Homework Statement Express \frac{d^{2}x}{dt^{2}} + \sin(x) = 0 In a system in terms of x' and y'. Homework EquationsThe Attempt at a Solution [/B] I seen this example: x^{\prime\prime\prime} = x^{\prime}(t)\cdot x(t) - 2t(x^{\prime\prime}(t))^{2} Where they then wrote: x^{\prime} =...

The second order ODE is, \begin{equation*} \frac{d^2 x}{dt^2} = -\omega^2_g \frac{dx}{dt} \end{equation*} I tried solving this by substitution of the second order derivative into a variable and transforming the equation into a second order polynomial, and I get the solution involving an...

Hello! I have this problem f'(x)=f(2-x) and I need to find f. This is what I did x -> 2-x f'(2-x)=f(2-2+x)=f(x) => f''(x)=f'(2-x)=f(x) => f''(x)=f(x) => ##f(x)=c_1e^x+c_2e^{-1}##. So, ##c_1e^x-c_2e^{-x}=c_1e^{2-x}+c_2e^{x-2}## => ##-c_2e^{-x}=c_1e^2e^{-x}## => ##-c_2=c_1e^2##. And, similarly...

Hi, I am learning ODE and I have some problems that confuse me. In the textbook I am reading, it explains that if we have a separable ODE: ##x'=h(t)g(x(t))## then ##x=k## is the only constant solution iff ##x## is a root of ##g##. Moreover, it says "all other non-constant solutions are separated...

Homework Statement So they want me to obtain the general solution for this ODE. Homework Equations I have managed to turn it into d^2y/dx^2=(y/x)^2. The Attempt at a Solution My question is, can I simply make d^2y/dx^2 into (dy/dx)^2, cancel the power of 2 from both sides of the equation...

Homework Statement Solve for the solution of the differential equation and use the method of variation of parameters. x`` - x = (e^t) + t Homework Equations [/B] W= (y2`y1)-(y2y1`) v1 = integral of ( g(t) (y1) ) / W v2 = integral of ( g(t) (y2) ) / W The Attempt at a Solution [/B] yc= c1...

xy'' + 2xy' - y = 0 Honestly no clue where to start, Wolfram Alpha gives a rather complex answer lol (http://www.wolframalpha.com/input/?i=xy%27%27%2B2xy%27-y%3D0)

Homework Statement Solve: y''+λ^2y = cos(λt), y(0) = 1, y'(π/λ) = 1 where t > 0 Homework EquationsThe Attempt at a Solution I start off by taking the Laplace transform of both sides. I get: L(y) = \frac{s}{(s^2+λ^2)^2}+\frac{sy(0)}{s^2+λ^2}+\frac{y'(0)} {s^2+λ^2} Now take the inverse...

Homework Statement Find the general solution y"+3y'+2y=0 Homework Equations y(t) =c_1e^r_1t + c_2e^r_2t The Attempt at a Solution a=1 b=3 c=2 r^2+3r+2=0 (r+2)(r+1)=0 r_1=-2 r_2=-1 General solution: y(t) =c_1e^(-2t)+c_2e^(-t)I was wondering if the order mattered. The answer in the book is...

Problem: y'=((x-1)/(x^2))*(y^2) , y(1)=1 . Find solutions satisfying the initial condition, and determine the intervals where they exist and where they are unique. Attempt at solution: Let f(x,y)=((x-1)/(x^2))*(y^2), which is continuous near any (x0,y0) provided x0≠0 so a solution with y(x0)=y0...

Homework Statement Find a particular solution to: (3x+2y+3)dx - (x+2y-1)dy = 0, y(-2) = 1 The answer to this problem as presented in the book ODE by Tenenbaum is the following: (2x+2y+1)(3x-2y+9)^4=-1. Homework Equations I will be shifting the origin to try to compute this problem. The...

Solve the ode $$(y-2x^2y)dx +xdy = 0$$ The equation is in exact form $$Q(x,y)dx+ P(x,y)dy =0$$ When I test for exactness it fails. Then I used the technique $$\frac{M_y-N_x}{N}$$ I get $u(x)=-2x$ as my integrating factor. But I end still end up with a non-exact d.e why is that...

Hi PF! Anyone have any ideas for a solution to this $$0 = F F''+\left.2F'\right.^2+ xF' + F$$ where primes denote derivatives with respect to ##x##. So far I have tried this $$0=\left( F F'\right)'+\left({xF}\right)'+\left.F'\right.^2$$ Which obviously failed. I also thought of this...