1
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1answer
43 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
0
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0answers
37 views

complex ordinary differential equation of a real variable

reading a paper i have found the following differential equation: f''[z] - (q^2)*f[z] == i*DiracDelta[z] here f[z] is a complex function of the real variable z ...
4
votes
1answer
58 views

Finding the period of the solution to $y'(x) = y(x) \cdot cos(x + y(x))$ with Fourier transform; how to interpret complex result?

A question elsewhere on this site asks about detecting the frequency of oscillations in a system defined by differential equations. The equation is $y'(x) = y(x) \cdot cos(x + y(x))$. The solution ...
4
votes
1answer
33 views

Unsure with second order complex differential equations

Solve $$y'' - 4y' + 5y = 0 $$ Where $y(0) = 0 \ , \ y'(0) = 2$. So I solve this as a second degree polynomial (no idea why) $$\frac{4 \pm \sqrt{16-20}}{2} = 2 \pm 2i$$ So the CASE III solution as ...
3
votes
2answers
72 views

Justifying an ODE's solution

In an introductory lesson into ODEs, in order to "semi-rigorously" justify the solution for e.g. : $(a)\ \ y'+y=0$ we proceed without an ansatz or guess solution (hence the "semi-rigour"): Let: ...
1
vote
2answers
25 views

ODE with complex char roots gives strange solutions

$y''-4y'+5y=0$ has char roots - $\{e^{(2+i)x},e^{(2-i)x}\}.$ So its solutions is $e^{2x}\cos(x), e^{2x}\sin(x).$ But when i plug, e.g., first of them into original eq. i get: $-4 e^{2x} cos(x) + 8 ...
1
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1answer
67 views

How do derive equivalent complex versions of linear differential equations.

I've done this before and have forgotten some of the details. I will try my best to re-derive it. Please help fill in the blanks. In my Acoustics book it says: An alternating force may be ...
1
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0answers
58 views

Can't match boundary conditions on a perturbation series solution to a non-linear ODE?

I'm trying to generate a naive perturbation series solution (with all associated secular terms included) to the Rayleigh equation: \begin{equation} \frac{d^2y}{dt^2} + y = \epsilon ...
1
vote
1answer
361 views

Solving the differential equation $y'' + 2y' + 2y = 0$ given constraints

How can I solve this initial value problem? $$ y'' + 2y' + 2y = 0,$$ given $y\,(\pi/4)=2$ and $y'(\pi/4)=0$. I've found $y(t)=e^{-t} \left(C_1\cos t + C_2\sin t \right)$ but I wasn't able to find ...
1
vote
1answer
193 views

How can you prove Euler's phase angle formula for differential equations?

How can you prove this formula: $C_1 e^{(\alpha + i\beta) t} + C_2 e^{(\alpha - i\beta)t}=Ke^{\alpha t}\cos {(\beta t + \phi)}$ This gives $x(t)$ in the second-order differential equation for an ...
0
votes
2answers
69 views

Two types of solution to the differential equation

I have read that we can have two solutions to the second order DE below, where $W$ and $W_p$ are constantants and $\psi$ is a function of $x$: $$\frac{d^2\psi}{dx^2} = -(W-W_p) \psi $$ (a) If we ...
4
votes
3answers
734 views

Calculate the $\int_0^{2\pi}\cos(mx)\cos(nx)dx$

I'm having trouble with this problem: Consider the integral: $$\tag 1\int_0^{2\pi}\cos(mx)\cos(nx)dx$$ a. Write $\cos(mx)$ and $\cos(nx)$ in terms of complex exponentials and compute ...
7
votes
3answers
286 views

Help understanding $e^{it}=\cos t+i\sin t$ by way of matrices and vector fields

I was brushing up on my complex arithmetic in preparation for a class in ODE's this semester and I found myself looking at Exercise 2.7.5 in Introduction to Complex Analysis for Engineers by Michael ...
1
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2answers
236 views

Apply Cauchy-Riemann equations on $f(z)=z+|z|$?

I am trying to check if the function $f(z)=z+|z|$ is analytic by using the Cauchy-Riemann equation. I made $z = x +jy$ and therefore $$f(z)= (x + jy) + \sqrt{x^2 + y^2}$$ put into $f(z) = u+ ...
2
votes
2answers
1k views

A-stability of Heun method for ODEs

I'm trying to determine the stability region of the Heun method for ODEs by using the equation $y' = ky$, where $k$ is a complex number, based on the method described here. If the Heun method is: ...
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2answers
117 views

Confused about characteristic equation of a linear ODE

Reading the Wikipedia page on linear ODEs, I've stumbled upon something I don't understand. In the section Homogeneous equations with constant coefficients, it says the following: If the ...
0
votes
2answers
2k views

Complex conjugate of function

I have a wavefunction $\psi(x,t)=Ae^{i(kx-\omega t)}+ Be^{-i(kx+\omega t)}$. $A$ and $B$ are complex constants. I am trying to find the probability density, so I need to find the product of $\psi$ ...
2
votes
1answer
227 views

Complex Numbers with the prey-predator equilibrium?

Consider a model (very similar to Lotka-volterra prey-predator -model, exception $h$). $$ \frac{dx}{dt} = h+x(\alpha -\beta y)$$ $$ \frac{dy}{dt} = -y(\gamma - \rho x). $$ Let's write this in ...
0
votes
1answer
163 views

Trigonometric Input to an First Order Differential Equation, Exponentials

In an ODE class, the differential equation is given $y' + ky = kq_e(t)$ where the input $q_e(t)$ is given as $cos \ \omega t$. The teacher "complexifies" the problem by using the real part of ...
8
votes
3answers
396 views

Proof for law of complex exponents using only differential equation

I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...