# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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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### 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 ...
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### 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 $\psi$ ...
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### 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 ...
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 ...
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 ...