# Tagged Questions

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### Solving $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by changing variables

Transform the differential equation $\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = 0$ by introducing new variables $x = u+v$ and $y=u-v$. then solve it. I which I could show ...
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### Particular solution of reducible to homogeneous equation

Verify that $y=x-5$ is a particular solution of the equation $$\frac{dy}{dx} = \frac{2y+6}{x+y+1}\ .$$ This is when $y'=1$ but this is not given as a condition in the question. How would you write the ...
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### Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
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### Variable substitution in second order PDE

Consider the the PDE $$A(x, y)\partial_{xx} u + B(x, y)\partial_{xy}u + C(x, y)\partial_{yy}u=h(x, y)$$ Now I want to make a variable substitution $\xi=f(x, y), \eta=g(x,y)$, so I can get $u$ as a ...
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### Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be differentiable and $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$.

The Assignment: Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be continuously partial differentiable and let $K \subset \mathbb{R}^n$ be compact and convex. Show $f$ is Lipschitz on $K$. A ...
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### Is speed a function of position?

Let $x$ be a smooth function from $[0,\infty)$ to $\mathbb{R}^n$ satisfying the following differential equation $x''(t) = f(x(t))$, where $f$ is a smooth function from $\mathbb{R}^n$ to itself. Then ...
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### Let U be open and $f: U \rightarrow \mathbb{R}$ be partial differentiable.

The Assignment: Let $U \subset \mathbb{R}^n$ be open and $f : U \rightarrow \mathbb{R}$ be partial differentiable and let all partial directional derivatives be continous function on $U$. Show ...
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### Partial differentiation for multi variables

A Candy company makes 2 types of candy A & B for which the average costs are 2 & 3euros per kg respectively. $Q_a$ & $Q_b$ (a & b subscript) are the kg that can be sold each week and ...
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### The Stable Manifold Theorem Applications

Definition: Let $\phi_t(x)$ be the flow of the nonlinear system $x'=f(x)$. The global stable manifold of $x'=f(x)$ at $0$ is defined by: $$W^s(0)=\bigcup_{t\leq 0}\phi_t(S)$$ Where $S$ is a ...
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### Calculus based question

For those who knows this stuff this is probably an easy question, but I don't know in general where I should consult to learn solution of this type of problems. So, beside providing solution/hint if ...
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### How to think when solving $3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$?

Solve this differential equation $$3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$$ Usually, when we get these problems, they tell us what variable change is smart to do and we just ...
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### existence of the solution of Neumann problem in $\mathbb{R}^3$

Let $D\subset \mathbb{R}^3$. Let $D$ be connected subset of $\mathbb{R}^3$. Show that there is not any solution of the system of equation \Delta u=f \text{ in } D, ...
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### How to compute time ordered Exponential?

So say you have a matrix dependent on a variable t: $$A(t)$$ how do you compute $$e^{A(t)}$$ It seems Sylvester's formula, my standard method of computing matrix exponentials can't be applied ...
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### How do I solve $F = \nabla\times G$ for $G$?

Given the vector valued function $F(x,y,z) = (xz,-yz,y)$ find $G$ such that $F = \nabla\times G$ I let $G(x,y,z) = (G_1,G_2,G_3)$ and expanded $\nabla \times G$ then equated the components to $F$ but ...
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### Solution in common for two differential equations

Consider: $E1: y''-4y'+4y=0$ Solution: $y(x)=c_1 e^{2x}+c_2 x e^{2x}$ $E2: y''-2ay'+(a^2-1)y=0$ Solution: $y(x)=c_1 e^{(a+1)x}+c_2 e^{(a-1)x}$ For what values of $a$, $E1$ and $E2$ have ...
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### Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Eulerâ€“Lagrange ...
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### Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
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### predictor-corrector method and stability

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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### Solution for a differential equation

If $x^n$ is a solution for the equation: $y''y'y=48x^9$, what is the value of $n$? I have the choices: 2, 1, 3, 4, or 6. I dont know how to resolve $y''y'y=0$, but maybe i dont need to do it. ...
For: $y''+ay'+by=0$ Given one of the solutions: $y(x)=e^x cos(x)$ Whats the value of $a+b$ ? I need some help with this, i dont know where to start.