# Tagged Questions

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### Decoupling system of two partial differential equations

If I have the following systems of PDE $$u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0,$$ where ...
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### Solving $u_{yy} + (2-x)u_y - 2xu = 1$

I want to solve the pde $$u_{yy} + (2-x)u_y - 2xu = 1$$ so if I treat $x$ in the coefficients as arbitrary but fixed it is equivalent to solving the ode $$y'' + (2-x) y' - 2x y = 1.$$ For the ...
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### Solving a PDE: basic first-order hyperbolic equation $u_t+cu_x=0$

So I have to solve the first-order hyperbolic equation $u_t+cu_x=0$ and $c$ as a constant. It is a PDE, since there is the time and spatial variable, but I'm overwhelmed by the maths given in books of ...
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### Obtaining characteristic v on Cauchy Problem

$(x-y)p+(y-x-z)q=z$ Find the integral surface which the curves it passes are $z=1$ and $x^2+y^2=1$ Here is my try. $$\frac{dx}{x-y}=\frac{dy}{y-x-z}=\frac{dz}{z}$$ So we have ...
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### How to solve Cauchy problem?

I'm new to this problem. Here is the question. $$(y+xz)z_x+(x+yz)z_y=z^2-1$$ Find the integral surface which the curves it passes are $y=1$ and $z=x^2$ By Lagrange system i found $u$ and $v$. We ...
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### Modified Symplectic Euler

Simple harmonic motion: $y'= -z$, $z'= f(y)$ and the modified Symplectic Euler equation are $$y'=-z+\frac {1}{2} hf(y)$$ $$y'=f(y)+\frac {1}{2} hf_y z$$ deduce that the coresponding approximate ...
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### A predictor-corrector method

A predictor-corrector method for the approximate solution of $y'=f(t,y)$ uses $$y_{n+1}-y_{n}=hf_{n} \tag P$$ as predictor and ...
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### Determining the expressions of the coefficients of the full Fourier series from the Complex Series

Let $l \gt 0$ be a positive real number, and $\phi:[-l,l]\rightarrow\mathbf{R}$ be the function defined by: $$\forall x\epsilon[-l,l], \phi(x)=e^x$$ (1) Calculate the coefficients of the Full ...
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### Elliptic PDE, uniqueness of solution

I'm considering a partial differential equation of the form $$\nabla^2 u + \mathbf{a}\cdot\nabla u = 0$$ with Dirichlet boundary conditions, where $\mathbf{a}$ is a (smooth, nonconstant) vector ...
I have this differential equation. $$Dc''+H=0$$ where the partial of the concentration is with respect to z the distance. H is the rate per unit volume of particles generated and D is the diffusion ...
i) IF $\frac{dy}{dt} = - \frac{∂H}{∂z}, \frac{dz}{dt}= \frac{∂H}{∂y}$ where H is a function of $y$ and $z$, show that $H(y,z)$ is constant in time. ii) Take a $H(y,z) =Ay^2 + 2Hyz + Bz^2$ where ...