Hi everyone,
The exam #3 review sheet is posted under “Classroom Resources/Exam Reviews”. Let me know if you have any questions or corrections.
Regards,
Prof. Reitz
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Differential Equations on Math.Stackexchange
- Solving the Differential Equation $ xy \dot{y} = y - 1 $I am given the differential equation: $$ xy \dot{y} = y - 1 $$ I have tried to solve it by separating the variables, but I have only come this far: $$ \int 1 + \frac{1}{y-1} dy = \int \frac{1}{x} dx = y + \log|y-1| = \log{|x|} + C $$ I am struggeling on how […]
- Solution of an IVPGiven $$\frac{dy}{dx}+p(x)y=0$$ its general solution is $$y=c\cdot\exp\left(-\int p(x)dx\right)$$ where $c\in\mathbb{R}$. Let $y(x_{0})=y_{0}$. My question is: how am I supposed to use the initial condition here? What I did was: let $\mu(x)=-\int p(x)dx$. Then $$y_{0}=c\cdot\exp(\mu(x_{0}))\Rightarrow c=y_{0}\cdot\exp(-\mu(x_{0}))$$ If $\exp(-\mu(x_{0}))=C$, we have $$y=C\cdot y_{0}\cdot\exp\left(-\int p(x)dx\right)$$ PS: In my assignment, I have to choose the correct answer. There's an […]
- How do I determine the stability of an equilibrium solution to a differential equation?According to the notes issued out by my class, "an equilibrium solution $N (t) ≡ c$ to a differential equation (DE) is stable if $N(t)$ stays close to $c$ whenever $N(t)$ is a solution to the same DE such that $N(0)$ is sufficiently close to $c$." What does this mean? I'm unsure as to how […]
- Do we have to set boundary at $0$?Hi all I have a fundamental question about heat equation, so far all the heat equation set up I have experienced asasume boundary at a boundary at $x=0$, (assuming it's one-dimensional). I'm wondering what may happen if I say boundary is $x=1$ for example (the domain is $x>1, t>0$). Will that change the solution? I […]
- How to convert a set of first order DEs to a single higher order DEI'm wondering how to convert a set of first order DEs into just a single DE. I've been looking at this for a while and I can't figure it out. I understand the opposite direction, a single DE into a system of first order but I just can't figure this out. Specifically I'm looking at […]
- Determining values of a parameter for which an initial value problem has a polynomial solutionGiven the following IVP: \begin{cases} \ddot{y}-3t\dot{y}-ky=0\\ y(0)=1; \; \dot{y}(0)=0 \end{cases} I want to determine the possible values of $k$ in order to obtain a polynomial solution; if $p(t)=a_0+a_1t+ \dots + t^n$ would be a solution; then $a_0=1; a_1=0$; so I'd get $p(t)=1+ a_2t^2+ \dots + t^n$ and I also have that: $$(-3n-k)t^n=0 \iff k=-3n$$ But […]
- I am struggling to connect the general transfer function formula for a second-order system with this specific question. [closed]Struggling with this transfer function question
- Polynomial solution of second-order differential equation$\ddot{y}-2t\dot{y}+3y=0 \quad(\ast)$ I want to investigate if there is any polynomial solution for the given differential equation Considering $y(t)=\sum_{n=0}^{\infty} a_nt^n$ and substituting above in $(\ast)$: \begin{align*} \sum_{n=2}^{\infty} n(n-1)a_nt^{n-2}-2\sum_{n=1}^{\infty}na_nt^{n}+3\sum_{n=0}^{\infty}a_nt^n=0 \\[10pt] \implies \sum_{n=0}^{\infty} (n+2)(n+1)a_{n+2}t^{n}-2\sum_{n=1}^{\infty}na_nt^{n}+3\sum_{n=0}^{\infty}a_nt^n=0 \\[10pt] \implies 2a_2+3a_0+ \sum_{n=1}^{\infty} \bigr[ (n+2)(n+1)a_{n+2}-2na_n+3a_n\bigr]t^{n}=0 \end{align*} If I assume that $a_0=0$, then $a_{2n}=0; n \in \mathbb{N}$, because $a_{n+2}=\frac{(2n-3)a_n}{(n+2)(n+1)}$. And if I let […]
- How to solve this difficult differential equation with Runge-Kutta methods?Given the following differential equation (it can be seen as a matrix differential equation): $$\begin{cases} x'=\dfrac{dx}{d\phi}=\dfrac{ \cos(\phi) }{ 2+\beta z-(1/x) \sin(\phi) } \\ z'=\dfrac{dz}{d\phi}=\dfrac{ \sin(\phi) }{ 2+\beta z-(1/x) \sin(\phi) } \\ x(\phi=0)=0; \\ z(\phi=0)=0 \end{cases} $$ How can I solve this using a Runge-Kutta method? $\phi$ exists in $[0,\pi/2]$. Thank you for the attention! I've […]
- Hypoteses about the stable manifold TheoremIn Perko's book, he proves the stable manifold theorem when the equilibrium point of the dynamical system is the origem. Can I use the same theorem for an equilibrium point that isn't the origem? Why?
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