# Tagged Questions

Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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### Arbitrary factors for the (modified) Mathieu equation

I am currently confronted with a physical equation that, after a fair amount of reworking, can be recast in the form of the modified Mathieu equation : y(x)'' - (a - 2q \cosh(2x)) y(...
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### Green's Functions: Solvable non homogeneous Sturm-Liouville with non homogeneous boundary conditions

I was just presented with this problem in my PDE Methods course which involves a non homogeneous Sturm-Liouville problem, which states as follows: Find the conditions under which the following SL ...
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### Stability of origin of dynamical system

Usually you can note some nice structure in the problem which enables construction of a nice Lyapunov function. But this one is just a monster. Maybe there is a trick I've missed? Investigate the ...
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### Reducing a higher order nonlinear ODE to a system of first order ODEs

The ODE that I am trying to reduce is: $$y''' + 4\,y'' + y' + 6\,y - 2y^{2} = 0$$ I start by letting $$y = y_1$$ $$y' = y_2$$ $$y'' = y_3$$ $$y''' = y_4 = 2y_1^2 - 4y_3 - y_2 - 6y_1$$ ...
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### Differential Equations which involve Infinite Series

The problem statement is as follows: Find the general solution for the following equation for $x(t)$. $$x''+ 9x = 2 + \sum_{n=1}^\infty \cos(nt)/n^3$$ I can't find anything about this in my ...
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### Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method

Solve $x(x-1)y''+6x^2y'+3y=0$ using Frobenius's Method I can't solve this ODE. How can I get first two term? and indicial equation is also very confusing. I can solve two term recurrence relation ...
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### Asymptotic Behavior of Differential Equation

physicist here. I'm studying some problems that involve the use of differential equations. The professor of the course has indicated that usually variable changes used to simplify the equations come ...
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### What type of equation is this?

Is this equation an ODE or PDE $$\frac{d^3u}{dx^3}−αxu=0, x∈R$$ The only thing given is $\int_R u(x) =\pi$ and $α>0$ is some constant. I have to find the solution using fourier ...
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### If $y'=\frac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2)$

If $y'=\dfrac{1}{x+1}$ and $y(0)=0$, find the value of $y(-2) = ?$ By integrating I am getting $$y = \ln (x+1)+C$$ I am stuck somewhat as it looks tricky from here. Any help ? Thanks!
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### Applied differential equation regarding water clocks

We have a water clock, the shape defined by $r=f(h)$, and the time marks on this water clock are equally spaced. We have to find f(h), and graph $h$ as a function of $r$, assuming the hole through ...
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### Writing ODE system with a complex variable

I'm looking at the system of ODEs: $$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$ I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first ...
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### A question about solving the nonlinear differential equation $\dot{x} = x(1-x)$

I am aware of the standard solution that makes use of partial fractions. However, I made the following manipulations, in order to be more rigorous with splitting up the differentials before ...
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### Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
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### Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
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So, I have (another) problem with differential equations (from an optimal control problem). I am trying to solve the following system of DEs (is this even a system?): $$\lambda'(t) = r \lambda(t) + ... 0answers 46 views ### Let \eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt then \eta is a C^1 function Consider the following problem. Suppose that a>0, r >0 and \xi:\mathbb R \to [o,\infty) is a C^2 which vanishes in the complement of the interval (-r,r). Also suppose that \xi(0)=\xi'(0)... 0answers 54 views ### What is the solution to the system \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}? I'm trying to solve the system$$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = kf_{...
$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...