# Tagged Questions

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

24 views

### Solving ODE by substitution. Where does $dy$ goes

When solving ODE by substitution, where does $dy$ goes from the following example? $$\left(1+\frac{sin(y)}{cos(y)}\right)dy=x dx$$ Let $u=-cos(y)$. Hence $du = sin(y)$, which results in the following: ...
35 views

### Cauchy-Riemann equations on $f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}$

Let $$f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}.$$ I need to show that the Cauchy-Riemann equations hold for $f$ in $0$ but $f$ is not (complex) differentiable in $0$....
42 views

48 views

### how do I solve following differential equation [closed]

$\frac{dx}{z(x+y)}=\frac{dy}{z(x-y)}=\frac{dz}{x^2+y^2}$ How do I solve this equation I am not getting it can someone help?please
88 views

### Bounds on a system of coupled ODEs

Suppose we have a $1$-dimensional differential inequality $$\frac{dx}{dt} \leq x - x^3$$ We can apply the Comparison principle to claim that if $y(t)$ is the solution to $\frac{dy}{dt} = y - y^3$, ...
43 views

### how do I solve the following differntial equations [closed]

1.$(D^2+3D+2)y=e^{2x}8x$ and $\frac{d^2y}{dx^2}-4x\frac{dy}{dx}+(4x^2-1)y=-3^{x^2}8*2x$ I am not getting any idea how to begin solving these problems
58 views

### Battle math equation [closed]

i am trying to code my own game and i am stuck at battle equation i will try to explain what i am trying to do as much as possible example of what i want to reach at the end http://2.bp.blogspot....
61 views

17 views

### Operator Splitting, Piecewise Function, ODE,PDE,Matlab code [duplicate]

$t_f=1$ $$y(t) = \begin{cases} e^{2t},& 0 \le t \le t_f/2 \\ 2\left(t-\frac{t_f}{2}\right)+e^{t_f} , & t_f/2 \le t \le t_f \end{cases}$$ How can i write this piecewise function in ...
26 views

### Orbit direction in Hamiltonian systems

The system $$\begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} y\\ x(1 + 2x^2) \end{pmatrix} =: f(x, y)$$ has the Hamiltonian $$H(x, y) = \frac{y^2 - x^2 - x^4}{2}$$ The orbits ...
26 views

### Brownian noise perturbing a differential equation

The following one-degree-of-freedom oscillator is given; $$\ddot{x}+kx=w(t),$$ where, $k>0$ and $w(.)$ is a Brownian noise perturbing the system. Assume we want to study boundedness of the ...
6k views

### Linear independence of function vectors and Wronskians

I am taking a course in ODE, and I got a homework question in which I am required to: Calculate the Wronskians of two function vectors (specifically $(t, 1)$ and $(t^{2}, 2t)$). Determine in what ...
33 views

### Finding all the eigenvalues and eigenfunctions for a BVP with an inequality condition

I am trying to find all the eigenvalues and eigenfunctions for the following boundary value problem \begin{eqnarray} \phi''(z) + \phi'(z) + \lambda \phi(z) &=& 0\\ \phi (0)&=& 0 \\ |\...
54 views

### Solving the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform, without missing solutions

I'm supposed to solve the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform and then explain if I got the most general solution. First of all, I don't know what "solve" means here ...
31 views

### Frobenius method solution of this nasty 2nd order Linear ODE

I've tried but can't get the solution of this ode by Frobenius method. $(x^2)y''-6y=0$ I tried with $y=\sum_{k=0}^{\infty}(a_k \cdot x^{(k+r)})$ where $a_k$ is coefficient. I can't find the ...
56 views

94 views

### I have a special solution for the Lane-Emden equation. Can I use it to find the general solution?

The general Lane-Emden equation is $$\ddot{y}+\frac{2\dot{y}}{x}+y^N=0$$ where $y(0)=1$ and $\dot{y}(0)=0$. If we eliminate the requirement that $y(0)=1$ there is a special solution for all real ...
157 views

59 views

### A general version of Gronwall's inequality

For the following $$|u(t)|^p\le C_1 \int_0^t |u(s)|^p\,ds+C_2$$ using Gronwall inequality, we have $$|u(t)|^p\le C_2(1+C_1 te^{C_1 t})$$ Now, my question is, for |u(t)|^p\le K_1 \int_0^t(1+|u(s)|^2)...
Verify that $y = −x^2$ is a solution for the equation $y' = x^3+ 2y/x-y^2/x$. Find the general solution for the equation. How would I go about solving this question?