**WeBWorK:** Assignments `14-InverseLaplaceTransforms,`

and `16-LaplaceIVP`

are due Tuesday, 5/24, at midnight.

**OpenLab:** OpenLab #3, Advice for the Future, is due on Tuesday, 5/21 (the day of the final exam).

"...how it differs from the rocks"

**WeBWorK:** Assignments `14-InverseLaplaceTransforms,`

and `16-LaplaceIVP`

are due Tuesday, 5/24, at midnight.

**OpenLab:** OpenLab #3, Advice for the Future, is due on Tuesday, 5/21 (the day of the final exam).

- Green's function for a differential operatorI am having a hard time with the following problem: $$ y\mapsto Ly=-\frac{d}{dx}(x\frac{d}{dx}y) $$ with boundary conditions $y(1)=0$ and $y^\prime(2)=0$. There are 2 things to do here: I have to determine a fundamental matrix for the differential equation $Ly=0$. I have to determine the green's function of L for these given boundary conditions and the […]
- Proving ODE does not have bounded solutionsConsider the equation \begin{equation} x'' + (a+b\cos t)x = 0 \end{equation} and let $u, v$ be solutions such that \begin{equation} u(0) = 1, u'(0) = 0; v(0) = 0, v'(0) = 1 \end{equation} Set $F(a, b) = u(2\pi) + v'(2\pi)$ and show that if $|F(a, b)| > 2$ then no solutions remain bounded for all […]
- Does $d(a L_t^\rho) \,dt = a L_t^\rho \ln(L_t/L_{t-1})$?I am trying to take a time derivative here and I am not sure if the following is true? Does anyone know what kind of time derivative rule was used here? Does $d(a L_t^\rho)\,dt = a L_t^\rho \ln(L_t/L_{t-1})$?
- How many linearly independent functions are solutions to an nth order linear differential equation?How many linearly independent real functions are solutions to an nth order linear differential equation? The answer is $n$ exactly. I would like to know(from a linear algebra perspective) why this is true. I am surprised that I could not find this question already asked here. I have checked many of them, so if you […]
- Given a initial value problem and a locally Lipschitz function, show it has unique solutionsGiven $f:\mathbb R^n \to \mathbb R^n$ is locally Lipschitz and $\Phi\in C^1 (\mathbb R^n, \mathbb R)$ with $\nabla\Phi(x)=f(x)$ for $x\in \mathbb R^n$, I would like to show that the initial value problem $$\begin{cases}\begin{split} y'(t)=-f(y(t))\\ y(0)=y_0 \end{split}\end{cases}$$ has locally unique solutions. I tried applying the P-L theorem, but it did not help. How do is this […]
- How to calculate the constants of the result of a differential equationI am trying unsuccesfully to solve this simple problem with Wolfram Alpha given an equation with constants. $$x(t) = c_2 \sin(t w) + c_1 \cos(t w) + \frac{g u}{w^2}$$ I want to calculate the values of $c_1$ and $c_2$ with the following initial values, $x(0) = x_0$, $x'(0) = 0$. This is indeed a simple […]
- Solving an ODE with a jump in the source termI am having a difficulty to derive the solution to the ODE \begin{align*} Ay'' - By = G(x) \end{align*} where \begin{align*} G(x) = \begin{cases} C \sin (kx) & |kx| < \pi \\ 0 & \text{Otherwise} \end{cases} \end{align*} The solution roughly reads \begin{align*} y = \begin{cases} p(\sin(kx) + kqe^{-\frac{\pi}{kq}} \sinh(\frac{x}{q})) & |kx| < \pi \\ pqk […]
- Having problems with drawing of a phase portrait of system of ODES in polar coordinatesI'm stuck with a problem which bothers me. It's about the following differential equation system: $$ x^{\prime}=y-(x^2+y^2-1)\cdot x\\ y^{\prime}=-x-(x^2+y^2-1)\cdot y $$ I have transformed the system in polar coordinates, which looks like: $$ r^\prime=-r\cdot(r^2-1)\\ \theta^\prime=-1 $$ Now to the part where I am desperate and don't know what to do: First of all, I want […]
- Third order non linear ODE - can anyone help me spot my error?I have this ode: $$y'''(1+y'^2) - 2y'y''^2 = 0 $$ I used the substitution: $y'=v,$ where $y'' = v'v$, and $y''' = v(v''v+v'^2)$, since $v=v(y)$ After plugging that in, I get the following equation: $$v(v''v+v'^2)(1+v^2)-2v^3v'^2$$ I used order reduction one more time: $v'=u, v''-u'u$, since $u=u(y)$ The equation transforms to: $$u'uv^2(v^2+1) - vu^2(v^2-1)$$ The solution […]
- Numerical method for Non smooth Ordinary Differential EquationI am considering the following "differential equation" \begin{equation*} \frac{dy}{dx} = \left\{ \begin{array}{ll} f_1(x)+g(x) & \quad x < 0 \\ f_2(x)+g(x) & \quad x \geq 0 \end{array} \right. \end{equation*} with $y(3)=0$,$f_1(0)=f_2(0)$, and $g(x)$ is a continuously differentiale function . Specifically $f_1$ and $f_2$ are two parts of a piecewise linear continuous function. Hence the right hand […]

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