Questions tagged [cauchy-problem]

Use this tag for questions about partial differential equations that satisfy certain conditions given on a hypersurface in the domain.

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A function such that any solution to the Cauchy problem has no entire solution in $\mathbb{R}^2$

I'm trying to figure out this problem: Find a smooth function $a(x, y)$ in $\mathbb{R}^2$ such that, for the equation of the form $$u_y + a(x, y)u_x = 0,$$ there does not exist any solution in the ...
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Non-unique solution of first order PDE

Question: $$\frac{\partial u}{\partial x} \frac{\partial u}{\partial y}=1 \qquad \qquad u=0 \; \text{ when } \; x+y=1$$ Find all possible solutions and state where each one exists. Attempt: Using ...
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Problem on the domain of the solution of a differential equation

Let $f:[0,\alpha]\to\mathbb{R}$ be a solution of the Cauchy problem: $\begin{cases} f'(t)=(f(t))^2+t \\ f(0)=0 \end{cases}$ The question is: prove that $\alpha<3$. It is clear that the problem ...
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Uniqueness of solutions of Cauchy problem with nonzero function

Consider the initial value problem $$y'(x)=f(x,y(x)), \\ y(x_0)=y_0,$$ where the function $f \colon D \to \mathbb R$ is defined and continuous on some open set $D \subseteq \mathbb R \times \mathbb R$ ...
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Solving a Cauchy Problem using Method of Characteristics

I came across a partial differential equation (IVP PDE) that I would like to solve: $$\{u_y+cos(ky)u_x=ax^2 | u(x,0)=0\}$$ This should be a quasilinear PDE, and is in the format of a Cauchy ...
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Extending three vectors to commuting vector fields

The following claim can be found in page 27 of Petersen's book "Riemannian Geometry": "... any three vectors can be extended to vector fields that commute." I have been unable to prove the statement,...
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Local uniqueness of solution for quasi linear PDE

I've a little troubles in proving local uniqueness of solution for Cauchy problems concerning quasilinear PDE's. It's a little bit boring, but I tried to be as clear as possible. Suppose $\Omega$ is ...