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
- Multiplying both sides by $dx$I have some questions about a proof I've seen that shows why we're allowed to "multiply both sides by $dx$" in separable ODEs. The proof goes like this: let $y=y(t)$ and $x=x(t)$ for some independent parameter $t$. Then, by the chain rule, we have $$\frac{dy}{dx}=\dfrac{dy/dt}{dx/dt}$$ which means that, in a separable ODE like $$\dfrac{dy}{dx}=y,$$ we […]
- Solving differential equatiom via complex numbers [closed]Solve $(D^3+D)y=\csc x$. Can it be done via $(D^3+D)z=e^{-ix}$? If yes/not,how/why?
- Finding the general solution to the given differential equationI want to find the solution to the following differential equation: $$y'=-\frac{4x+3y-1}{3x+4y+1}$$ I use the substitutions $x=s+1$ and $y=u-1$ to obtain $$\frac{dy}{dx}=\frac{du}{ds}=\frac{-4(s+1)-3(u-1)+1}{3(s+1)+4(u-1)+1}=\frac{-4s-3u}{3s+4u}.$$ If I now further substitute $z=\frac{u}{s}$ this yields $$\frac{du}{ds}=s\frac{dz}{ds}+z=\frac{-4s-3zs}{3s+4zs}$$ $$\frac{dz}{ds}=\frac{1}{s}\frac{-4s-3zs-3zs-4sz^2}{3s+4zs}=\frac{-4-6z-4z^2}{s(3+4z)}$$ This is now a seperable equation so I can integrate both sides $$-\int \frac{3+4z}{4+6z+4z^2}dz = \int \frac{1}{s}ds.$$ By substituting $w=4z^2+6z+4$ I can solve […]
- Finding the value of $k$ if the differential function is an identity?Question $$\frac{\mathrm d^3x}{\mathrm dy^3} + {\left(\frac{\mathrm dy}{\mathrm dx}\right)}^{-4}\frac{\mathrm d^3y}{\mathrm dx^3} - k {\left(\frac{\mathrm dy}{\mathrm dx}\right)}^{-5}{\left(\frac{\mathrm d^2y}{\mathrm dx^2}\right)}^{2}=0$$ is an identity, find the value of $k$. My approach I know identity means this equation should be satisfied for any function. One easy way to do it is assume $y=x$ or any other simple function. But I […]
- Analyse the behaviour of Van der Pol oscillator using Finite Difference Method with Newton iteration method in MATLAB.I'm trying to analyze the behavior of van der Pol oscillator using Finite Difference Method with Newton iteration method in MATLAB. I have try to solve the problem but getting singular Jacobian matrix. Can any one suggest how to proceed to get it's true behavior (oscillation). Here is MATLAB code that I have used for. […]
- Can we find a neighbourhood around a singular point such that the given expression is undefined in the neighbourhood too?I was solving the following differential equation: $$x^2y''+(1+x)y'+3y = 0$$ I tried to express the equation in the form: $$y''+p(x)y'+q(x)y = 0$$ where $p(x) = \frac{1+x}{x^2}$ and $q(x) = \frac{3}{x^2}$. The exclusion of the point $x=0$ for the second equation is understood but my question is that does there exist a $\delta$-neighborhood of $0$, such […]
- How to solve $\frac{dy}{dx}=\frac{x+y+4}{x-y-6}$? [duplicate]I have started learning ODE and I couldn't solve this exercise: $$\frac{dy}{dx}=\frac{x+y+4}{x-y-6}$$ I am a beginner and only learned separation of variables and how to solve homogeneous equations by using $y=ux$ but I don't think this problem can be solved by any of these. I tried to use sub $x-y=u$ but I didn't get anything […]
- How can I adjust the modified Bessel functions for a scalar co-efficient of y'? [closed]I have a 2nd order ODE of the form: $$ t^2y'' + 3ty' -t^2y =0$$ The modified Bessel functions $I_{\nu}$ and $K_{\nu}$ are defined as the solutions to the modified Bessel equation: $$t^2y'' + ty' - (t^2 + \nu^2)y =0$$These equations are clearly very similar. Is there a way to modify the Bessel equations to […]
- On extensions of solutions to ODEs.I'm struggling to understand extensions of solutions for ODEs. Consider the following initial value problem: $$x'(t) = f(t, x), \quad x(t_0) = x_0.$$ Assume that $f$ is continuous and bounded on its domain $\Omega$. Typically, maximal solutions are constructed by defining a function that agrees on intersections, which is possible when we have uniqueness. However, […]
- How to solve an heat kernel-type equation [closed]I was wondering if there is an elegant way to solve the following heat kernel equation, $$\left[2 \omega p-a i p^{2}-a i p \frac{\partial}{\partial p}\right] K(p,k,\omega,s)+i \frac{\partial}{\partial s} K(p,k,\omega,s)=i \delta(s) \delta\left(p-k\right).$$
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