Hi everyone,
The grades for Exam 3 can be found under Dashboard/OpenLab Gradebook.
The exams will be returned on Tuesday.
Best,
Prof. Reitz
"...how it differs from the rocks"
Differential Equations on Math.Stackexchange
- Intuition behind dampening as a rotation in the complex planeI'm currently building my intuition behind the meaning of the variables within the general second-order damped harmonic oscillator frequently taught in engineering: $$ \ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2 x = 0 $$ Assuming an underdamped case, the eigenvalues are as follows: $$ \lambda = -\zeta w_n \pm w_n \sqrt{1 - \zeta^2} i $$ As such, […]
- Second-order differential equation of the form $y''y' = f(x,y,y')$ [closed]Can anyone provide examples of physical situations or mathematical problems where differential equations of the form $y''y' = f(x,y,y')$ are useful?
- Differential equation LagrangeI have a problem solving this differential equation: $$ y = x(y')^2 - y' $$ Is anyone able to work out for me how to do this because I am at a loss? I would be very grateful for your help! Thanks.
- Prove that two integrating factors define a solution.I've been toiling away at this proof problem from Chapter 2.4 ending exercises of Differential Equations 3rd ed by Shepley L. Ross, but to no avail. Show that if $\mu (x, y)$ and $v(x, y)$ are integrating factors of $$ M(x, y)\ dx + N(x, y)\ dy = 0 $$ such that $\mu (x, y) […]
- How to solve $2xy\frac{dy}{dx}+(1+x)y^2=e^x$?$2xy\frac{dy}{dx}+(1+x)y^2=e^x,\qquad y(1)=\sqrt{\frac{e^2+2}{2e}}$ Performing the substitution $y(x) = (z(x))^{1/2}$ what is the general solution and what is the value of $y(2)$ After doing some calculus I arrive to that final expression I'm guess I'm doing something wrong because the expression seems too complicated to integrate
- Are the Particular Integral and Complementary Function orthogonal?I am a physics student so please forgive my lack of rigour. Suppose I have a Linear, Inhomogenous O.D.E given by Ly(x) = f(x). I may construct a solution from the Greens function, G(x,z), by integrating the driving function f(x) with G(x,z). My first question is whether this solution is equal to C.F.+P.I. or just […]
- Periodic perturbation of ODERecently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.5, the author mentioned that the following is well-known in the theory of ODEs. If $x_0$ is an asymptotically stable equilibrium point of an autonomous system $\mathrm{d}x/\mathrm{d}t=F(x)$ and $f(t)$ is $\theta$-periodic, then for sufficiently small $\epsilon$ the system $\mathrm{d}x/\mathrm{d}t=F(x)+\epsilon f(t)$ has a […]
- Uniqueness of solutions of high-order differential equationsI was quite confused about Converse to Wronskian Condition ODEs " $y(x_0) = y'(x_0) = \cdots = y^{(n-1)}(x_0) = 0$ then the result follows from a contradiction of the uniqueness theorem, since y=0 is also a solution. " When defining solutions to higher-order differential equations, do we assume that we only consider solutions that satisfy […]
- Need help to solve Sturm-Liouville ODE of form $\frac{dy}{dx}\Big(p(x)\frac{dy}{dx}\Big) + (\lambda)w(x)y = 0$The first bit of information is just to provide context for where the equation I need help with came from, if you are uninterested, you can just skip to equation (2). While solving a Poisson's Equation with Dirichlet BCs using a Laplacian in spherical coordinates (with $\theta$ and $\phi$ being our azimuthal and polar angles) […]
- Why is the uniqueness theorem for 1st order differential equations involve taking a partial dervative?I'm looking for intuition on why the uniqueness theorem for 1st order differential equations on initial value problems involves looking at a partial derivative. Say f(x,y), the theorem says if f is continuous and $f_y$ is continuous, then the solution through some open rectangle containing the initial point is unique. How does $f_y$ being continuous […]
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