# Questions tagged [cauchy-problem]

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

112 questions
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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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### Solving a Cauchy problem

Given the equation $$y' = e^t \sqrt[3]{y^2}$$ (a) consider the related Cauchy problem with $y(t_0)=y_0$. What $P(t_0,y_0)$ ensures the problem has a unic solution? (b) find the general integral of ...
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### Ordinary Differential Equation (1°ord.), positive or negative constant when solving Cauchy problem?

I have this easy ODE: EDIT: $$y'(x)=2\sqrt{y(x)}, \\y(0)=1$$ (OLD: $y' = (x - 2)/2;$ $y(0) = 1$) The general integral/solution is $\sqrt{y} = x + c$, so $y= (x + c)^2$ Substituting the initial ...
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### Triple-valued solution to Riemann problem - Profile of the bulge

I am studying conservation laws and hyperbolic systems, particularly, Burgers' equation and shocks, and have a doubt at pages 48/49 of the book Numerical Methods for Conservation Laws by R.J. LeVeque (...
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### Studying a solution of a Cauchy problem

Consider the Cauchy problem $y' = \frac{t+2}{t^2+y^2}$, $y(0)=1$, study the behavior of its solutions, if it possible, when $t\to +\infty$. By Cauchy-Lipschitz there exists a unique local solution; ...
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### Problem with first order linear ODE formula.

I'm working at this simple linear first order ODE: $$y'+y=e^{x}$$ Rewriting as: $$y'=-y+e^{x}$$ I want to apply the formula: $$y(t)=e^{\int a(t)dt}\int e^{-\int a(t)dt}b(t)dt$$ Where, in this ...
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### How do I solve Cauchy problem for $y''-2y'-3y = e^{4x}$?
I have one: $$y''-2y'-3y = e^{4x} \quad y(0) = 1 \quad y'(0) = 0$$ I've found the solution as a sum of general solution and particular one: $$y(x) = C_1e^{-x}+C_2e^{3x}+\frac{1}{5}e^{4x}$$ ...
Consider the Cauchy problem $yu_x-xu_y=0$ where $u=g$ on $S=${$(x,y): x+y=1, x>1$} Prove that the Cauchy problem has a unique solution in a neighbourhood of $S$ for every differentiable function \$...