# Tagged Questions

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### Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
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### checking the solution of PDE

Let $u(x)$ be an entire solution of $\Delta u = 1$ on $R^{n}$, $n>1$. If $u(x)$ is also convex, that is $(D^{2}u(x))$ is non-negative definite for all $x$. Then $u(x)$ is given be a quadratic ...
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### Fourier transform help for solving $u_t+u_{xxxx}+u_{xx}=0$

I just started to learn a little bit of fourier analysis in solving PDEs. I want to find a solution $u(x,t)$ to $u_t+u_{xxxx}+u_{xx}=0$. My attempt: Applying the fourier transform to both sides gives ...
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### Estimation of solution to $u_t=u_{xx}+x^3u_x$ using integrals

Let $$u_t=u_{xx}+x^3u_x$$ With: $$u(0,x)=u_0(x)$$ $$u(t,0)=u(t,l)=0$$ Find an energy approximation of $u$ on $(0,T) \times(0,l)$. By multiplting by $u$ we get: ...
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### Weak solution Boundary Value problem

I have to prove that the following problem $$(P) \begin{cases} -u''-u=1\,\,\,\,\,\,\,\,\text{if}\,\,\, x\in(0,\pi)\\ u(0)=u(\pi)=0 \end{cases}$$ doesn't admit weak solutions. I'm proceeding by ...
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### To solve a non-homogeneous linear PDE

To solve a non-homogeneous linear PDE $\displaystyle \frac{\partial^2 z}{\partial x^2}+\frac{\partial^2 z}{\partial x \, \partial y}+\frac{\partial z}{\partial y}-z=e^{-x}$ My Attempt: Putting ...
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### To Solve $\displaystyle p(p^2+1)+(b-z)q=0$ using Charpit's Method

p = ∂z/∂x q = ∂z/∂y My Attempt: $\displaystyle \frac{dx}{-3p^2-1}=\frac{dy}{b-z}=\frac{dz}{-p(3p^2+1)-q(b-z)}=\frac{dp}{-pq}=\frac{dq}{-q^2}$ Now, I can't think ...
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### To Solve $\displaystyle pxy+pq+qy=yz$ using Charpit's Method

p = ∂z/∂x q = ∂z/∂y My Attempt: $\displaystyle \frac{dx}{-py-q}=\frac{dy}{-p-y}=\frac{dz}{-p^2y+qy}=\frac{dp}{0}=\frac{dq}{px+q+qy}$ Now, $\displaystyle dp=0$ ...
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### Solving PDEs using Charpit's Method

To Solve : $\displaystyle 1+p^2=qz$ I have solved this equation till auxiliary equation: $\displaystyle \frac{dp}{-pq}=\frac{dq}{-q^2}=\frac{dz}{2p^2-qz}=\frac{dx}{2p}=\frac{dy}{z}$ ...
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### Solve initial value problem (C.S.I.R)?

The initial value problem is $$\frac{\partial u}{\partial t} +x\frac{\partial v}{\partial x} = x, \ \ 0 \leq x \leq 1, \ \ t > 0 \ \ and$$ $$u(x,0) = 2x \ \ has$$ a unique solution $u(x,t$ ...
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### Solving $\displaystyle p(1-q^2)=q(1-z)$

To Solve: $\displaystyle \frac{\partial z}{\partial x} \left \{1-\left(\frac{\partial z}{\partial y}\right)^2\right \}=\frac{\partial z}{\partial y}(1-z)$ My Attempt: Assume $z$ is a function of ...
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### Form a PDE by eliminating the arbitrary function

Form a PDE by eliminating the arbitrary function (i) $\displaystyle F(xy+z^2,x+y+z)=0$ (ii) $\displaystyle F(x+y+z,x^2+y^2+z^2)=0$ How do I proceed for any of these?
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### Doubt on solution of PDE

To Solve: $\displaystyle (x^2-y^2-z^2)\frac{\partial z}{\partial x}+2xy\frac{\partial z}{\partial y}=2xz$ Subsidiary equation: $\displaystyle \frac{dx}{x^2-y^2-z^2}=\frac{dy}{2xy}=\frac{dz}{2xz}$ ...
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### To Solve a PDE by direct integration

$\displaystyle \frac{\partial^2 z}{\partial x^2}=a^2z$, given that when $\displaystyle x=0, \frac{\partial z}{\partial x}=a\sin y$ and $\displaystyle \frac{\partial z}{\partial y}=0$ The solution ...
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### How do i solve this simple PDE? $\partial_x u + \partial_y u =0$

Can you please give me a hint on how to start this differential equation? $u$ is a two variable real function. $$\partial_x u + \partial_y u =0$$ Thank you in advance!
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### Solution of a PDE

Solve $\displaystyle \frac{\partial^2 z}{\partial x^2}+z=0$, given that when $x=0$, $z=e^y$ and $\displaystyle \frac{\partial z}{\partial x}=1$. My Attempt: Integrating w.r.t x twice (keeping y ...
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### Solve this PDE using a change of variable.

Solve the following PDE using a change of variable: $$\alpha^2 \dfrac{\partial^2 z}{\partial x^2} - \beta^2 \dfrac{\partial^2 z}{\partial y^2} = 0$$ This is my attemp: Let the following change of ...
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### Using Fourier Transforms to Solve the Heat Equation PDE In Infinite Three Dimensions

Problem: Using Fourier transforms, solve for $u(x,y,z,t)$, where $$u_t=D\nabla^2 u$$ $$-\infty<x,y,z<\infty,t>0$$ $$D>0, u(x,y,z,0)=f(x)f(y)f(z)$$ and $u\rightarrow 0$ as ...
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### Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...