Hi everyone,

The review sheet for Exam #2 (which will take place on Tuesday, April 2nd) is posted on the Exam Reviews page. Let me know if you have any questions.

Best regards,

Prof. Reitz

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"...how it differs from the rocks"

Hi everyone,

The review sheet for Exam #2 (which will take place on Tuesday, April 2nd) is posted on the Exam Reviews page. Let me know if you have any questions.

Best regards,

Prof. Reitz

- How is it possible to demonstrate the necessary system to find particular solutions in LDE of order 2When when want to find particular solutions to a LDE of order 2 like : $$ y'' + ay' + by = f(x) $$ I see in a lot of source that says that after finding two solutions for the homogeneous equations $y_1, y_2$, we want to find two functions $\lambda(x), \mu(x)$ that satisfies the […]
- In what time $\tau$ will the particle reach the point $x=0$?A particle of mass $m$ capable of moving along the $x$-axis, is acted upon by a force $F(x) = -\frac{k}{x^3}$. At the initial time moment $t=0$, the particle is at the point $x=x_0>0$, and its velocity is $\dot{x}(0)=0$. In what time $\tau$ will the particle reach the point $x=0$? My attempt: We basically have to […]
- Finding parameter values for which a system has closed orbitsMy question is on Exercise 7.3.8 of Chaos and Nonlinear Dynamics (2nd ed) by Strogatz: 7.3.8. Recall the system $\dot{r} = r(1-r^2) + \mu r \cos \theta, \; \dot{\theta} = 1$ of Example 7.3.1. Using the computer, plot the phase portrait for various values of $\mu > 0$. Is there a critical value $\mu_c$ at […]
- Checking if a first order differential equation follows certain initial conditionsI am trying to integrate the following first-order ordinary differential equation and confirming whether it is possible such that $t(r) = 0$ when $r = 0$. $$\int \frac{dr}{r^2((a - r)^2 + b)} = \int c dt$$ From computationally integrating the above equation, I have obtained $t(r)$ as the following. $$t(r) = -\frac{a \log(a^2 - 2 […]
- Proving solutions of $y''+p(x)y'+q(x)y=0$ to be linearly independentWhen studying Elementary Differential Equations by William, I found trouble understanding Theorem 5.1.5 It says the two solutions are linearly independent iff their Wronskian is never zero, but I think they can still be linearly independent even if the Wronskian is zero for some $x$. In the proof, when $W(x_0)=0$, Theorem 5.1.4 is used to […]
- Numerical Analysis of Differential Equation with Boundary ConditionsIn a thick-walled cylindrical pipeline, a hot liquid flows with a constant temperature $T_i = 450^\circ$C. The cylinder wall has an inner radius of 1 length unit and an outer radius of 2 length units. The temperature distribution $T(r)$ within the cylinder wall ($1 \leq r \leq 2$) is determined by the differential equation: $$ […]
- find time needed for a given displacement where velocity is nonlinear in timeA velocity v is obtained as a function of time t from a nonlinear ordinary differential equation and thus the inverse function t(v) can't be written down analytically in general. How to find out the time T needed to reach the displacement s, which is the integral of v(t) w.r.t t from 0 to T? […]
- Necessary condition for exponential stability of an LTV systemThis is a homework problem from my adaptive control course: Consider the IVP $\dot x(t) = -u(t)^2x(t)$ with $x(0) = x_0$. Suppose the system is exponentially stable. Show that there exist some $\epsilon, T > 0$ such that $$ \int_t^{t + T} u(\tau)^2d\tau \geqslant \epsilon T $$ for any $t > 0$. Below is my […]
- what more can I say using Sturm's comparison theorem?Can I use Sturm's comparison theorem to say something about the average? Let $f(t)$ be a continuous function such that $\lim_{t \to \infty } f(t) = \infty$. Let us consider the following function $$ \ddot{x}(t) + f^2(t) x(t) = 0. $$ Let $C$ be a constant. Then, we can find $T$ such that $C^2 < […]
- How to derive an inequality for the solution of a differential equation based on a differential inequality?Suppose that the following differential inequality holds: $$ \dot{y}(t) \leq -c\cdot y(t) + \lambda \ \ (1)$$ with $c,\lambda > 0 $ positive constants. Now let $\rho = \lambda /c > 0$, I'm wondering how the following is concluded from equation (1): $$ y(t) \leq \rho + y(0)\cdot e^{-c\cdot t} - \rho\cdot e^{-c\cdot t} \ […]

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