WeBWorK: Assignments 7-Trench-EulersMethod and 7b-Trench-ImprovedEulersMethod will be due on Tuesday 3/12 at midnight.
OpenLab: none
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Differential Equations on Math.Stackexchange
- How to use the substitution method to end up with a second-order linear ODE with non-constant coefficients and solve that ODE?We face a system characterized by the following matrix : $$ \begin{cases} \dot{x} = -2tx + t^2y \\ \dot{y} = t^3x + 4ty \end{cases} \quad\Longrightarrow\quad A = \begin{pmatrix} -2t & t^2 \\ t^3 & 4t \end{pmatrix} $$ Since $A$ is time-dependent, the usual techniques for a by-hand/no-computer resolution do not work as is, but they […]
- Some interesting ODEs I am contemplating (just two).The ODE is as follows: $$y''+1/y'=y,y(0)=0,y'(0)=1$$ $$y''+1/y=y', y(0)=1,y'(0)=0$$ The first one I multiply by $y'$ to get the following ODE: $y'=\sqrt{y^2-2x}$, don't know how to resolve this ODE though. The second ODE, again I should do the same trick of multiplying the equation by $y'$ to get: $\frac{1}{2}(y')^2+\ln(y)=\int^x (y')^2 dx$. Here I guess I could […]
- Probability density function that satisfies an ODELet $k_1
- integrate the following differential equation xy''-tan(y')=0I need solve this problem but i'm really having problems What i had done: $$xy''=tan(y')$$ $$\text{use: }u=y'$$ $$xu'=tan(u)$$ $$\text{We will find } u=arcsin(Cx)$$ So I thought of replacing the u in $y'=arcsin(Cx)$ $$\frac{dy}{dx}=arcsin(Cx)$$ $$dy=arcsin(Cx)dx$$ And after that integrate to find y. Does this make sense for this exercise? I don't know if the method I […]
- How do I solve an initial condition, second order ODE with the Fourier transform? y'' + 20y' + 6000y = 0, y(0) = 1, y'(0) = 0? Help pls. [closed]I am struggling to solve this problem with the Fourier Transform??? I transform every term and factor out F(w), but from there the other side of the equation = 0, and I don't know how to proceed. Please help me.
- Dynamical System Local Asymptotic Stability with Nontrivial KernelI’m trying to investigate the local asymptotic stability of critical points in a certain autonomous system. Specifically, we’re looking at trajectories satisfying $\dot{x} = f(x)$ for a non-linear $f$ on a Banach space $V$. The issue is that the critical points $x_0$ such that $f(x_0) = 0$ are not necessarily isolated. Given a critical point […]
- Solving Nonhomogenous Systems of ODE's with Linear Independence Issues via Undetermined Coefficientsbeen looking to better understand how to find the solution to the system $\vec{x}'=\begin{pmatrix}1&-26\\1&-1\end{pmatrix}\vec{x}+\begin{pmatrix}4\\0\end{pmatrix}cos(5t)$ using the method of undetermined coefficients, now given that the associated homogenous system $\vec{x}'=A\vec{x}$ has eigenvalues of $\lambda=\pm 5i$ I know the regular guess of $\vec{x}_p=\vec{a}cos(5t)+\vec{b}sin(5t)$ won't work, I've also read somewhere that when this happens we try the guess $\vec{x}_p=\vec{a}cos(5t)+\vec{b}sin(5t)+\vec{c}tcos(5t)+\vec{d}tsin(5t)$ […]
- Mellin transform and boundary conditionLet $k>0$, $f\in C[0,+\infty)$. Consider the problem $$ -u''(r)-\frac 1 r u'(r)+\frac 1 {k^2}u(r)=f(r), r>0,\tag{1} $$ $$u(0)=0, u(+\infty)=0. \tag{2}$$ By the Green function method one can easily obtain $$ u(x)=\int^{x}_0 \left(\frac s r\right)^ksf(s)ds+\int^{\infty}_x \left(\frac r s\right)^ksf(s)ds, \tag{3}$$ and see that $u(0)=0$. Use on the other hand the Mellin transform $u_*(p)=\int^\infty_0 r^{p-1}u(r)dr$. We have (formally) $$ […]
- Questions regarding the proof of the Rayleigh’s equation has at least one limit cycleThis comes from the book Classical Mechanics by R.Douglas.Gregory In page 207, we are given an equation known as The Rayleigh equation $$x'=y$$ $$y'=-x-\epsilon y(y^2 -1)$$ and the book aims to prove that The Rayleigh’s equation has at least one limit cycle. The book states that the above system of equations in polar form is […]
- Stuck on a linear system of DEs by Laplace transform$x_1''-2x_2'+3x=0$, $x_2''+2x_1'+3x_2=0$ where $x_1(0)=4,x_1'(0)=0,x_2(0)=0,x_2'(0)=0$ Applied Laplace transform to both equations $4s=(s^2+3)X_1-2sX_2 ...(1)$ $8=(s^2+3)X_2+2sX_1$ can yall guide me to where to move forward from here, please.
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