Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [cauchy-problem]

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

7
votes
0answers
188 views

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 ...
6
votes
0answers
68 views

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 ...
5
votes
2answers
313 views

Cauchy differential equation

I'm trying to resolve this cauchy problem: $ y'=2y+1$ such as $y(0)=1$ the general integral for the differential equation is $\frac{1}{2}(e^{2x+2c_1}-1)$ for $y(0)=1$ : $y(0)=\frac{1}{2}(e^{2c_1}-...
4
votes
1answer
47 views

Demonstrate that solution of differential equation is bounded

Let $b(t) \in C^1([0,+\infty))$. I have to find a formula for the solution of this Cauchy's problem: $$ \left\{\begin{aligned} x''(t)+x(t)&=b(t) \\ x(0)&=x_0\\ x'(0)&=x_1 \end{aligned}\...
4
votes
2answers
648 views

Unique solution of Cauchy problem

Let , $a,b,c,d \in \Bbb R$ such that $c^2+d^2 \not =0$. Then the Cauchy problem $au_x+bu_y=e^{x+y}$ , $x,y\in \Bbb R$ with $u(x,y)=0 $ on $cx+dy=0$ has a unique solution if (A) $ac+bd \not=0$. (B) $...
4
votes
1answer
135 views

Finding when Cauchy data is characteristic

I have been stuck on the following problem: Consider the Cauchy problem \begin{equation} \frac{\partial^2 u}{\partial x_1 \partial x_2} - 4\frac{\partial^2 u}{\partial x_3^2} + \frac{\partial u}{...
4
votes
1answer
42 views

Limit of solution of Cauchy problem

I have the following Cauchy problem: \begin{align} y'(t) = \arctan(t^3(y-1)) \\ y(0) = \alpha \end{align} I want to study the limit of the solution on the boundary. This is what I have done so far:...
4
votes
1answer
89 views

Continuity of mild solution

This is a question on the proof of Theorem 6.1.2, Pazy's book Semigroups of Linear Operators and Applications to Partial Differential Equations. Also the title might not be too accurate, as my ...
3
votes
1answer
163 views

Find equation for solution of differential equation.

Please help me to solve this problem: Differential equation $y''+p(x)y=0$ has nonzero solution $f(x)$. Find the equation for function $z(x)=\frac{f'(x)}{f(x)}$. My ideas: I only see that $z(x) = \...
3
votes
4answers
79 views

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 ...
3
votes
1answer
75 views

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 $D \subseteq \mathbb R \times \mathbb R$ ...
3
votes
2answers
139 views

How to solve Cauchy problem for $y' + y\cos x = e^{-\sin x} \quad y(0) = 1$?

I have: $$y' + y\cos x = e^{-\sin x} \quad y(0) = 1$$ Applying Lagrange's method I got: $$y' + y\cos x = 0 \\ \frac{dy}{dx} = -y\cos x \\ \int\frac{dy}{y} = -\int \cos x \, dx \\ \ln |y| = -\sin x +...
3
votes
2answers
76 views

Doubt about Cauchy-Lipshitz theorem use

I'll show my doubt with the Cauchy problem $\begin{cases} y'=1+y^2 \\ y(0)=0 \end{cases}$. I know the solution is $y(t)=tg(t)$, but let's say I don't know the explicit solution. If I look at $1+y^2$, ...
3
votes
1answer
420 views

Find the solution of the Cauchy problem $u_t-uu_x=0$.

For the Cauchy problem $u_t-uu_x=0, ~x\in \mathbb{R}, t>0$ with $u(x,0)=x,~x\in \mathbb{R} $, which of the following statements is true? the solution $u$ exists for all $t>0$. the ...
3
votes
1answer
81 views

Cauchy problem using Duhamel's Principle

Consider the Cauchy problem $$\left\{ \begin{array}{l l} u_{tt} - u_{xx} = e^{-t}\sin x & \quad \mbox{$x \in \mathbb{R}, t>0$,} \\ \quad u(x,0) = 0, \\ \quad u_t(x,0) = 0. \end{array} \right....
3
votes
1answer
329 views

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 ...
3
votes
0answers
56 views

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,...
2
votes
2answers
50 views

Solving a Cauchy problem, differential equation

I have the following Cauchy problem \begin{cases} y'(x) + \frac{1}{x^2-1}y(x) = \sqrt{x+1} \\ y(0) = 0 \end{cases} I proceed by finding $e^{A(x)} $ where $A(x)$ is the primitive of $a(x)= \frac{1}{...
2
votes
2answers
157 views

Explanation to PDE $u_{tt}-c^2u_{xx}=0$ solution

Exercise: Solve the Cauchy problem $u_{tt}-c^2u_{xx}=0$ with conditions $u(x,0)=g(x)$ and $u_t(x,0)=h(x)$, where $g(x)=0,\ h(x)=\begin{cases} 0,\ x<0 \\ 1,\ x\ge0 \end{cases}$. Please ignore this ...
2
votes
1answer
40 views

Solutions of a PDE problem given in the Riemann invariant diagonal form

First, let's say we have a Cauchy problem: $$ (1) \hspace{0.5cm} u_t (x,t)+ div f(u(x,t))=0, $$ where the initial condition is given with $u(x,0)=u_0(x)$, $x \in A \subseteq \mathbb{R}^d, d \geq 1$,...
2
votes
1answer
39 views

Infinitely many solutions for a first order Cauchy problem.

Is this correct that the following Cauchy problem has infinitely many solutions? ‎\begin{cases}‎ ‎xu_t+u_x=0 \\‎ ‎u(x,0)=\cos x‎ ‎\end{cases}‎ Using the method of characteristics it is obvious ...
2
votes
1answer
50 views

Show that problem is well defined for each time

We have the Cauchy problem of the equation $u_t+xu_x=xu, x \in \mathbb{R}, 0<t<\infty$ with some given smooth ($C^1$) function $g$ as initial value. I want to check if the problem is well ...
2
votes
1answer
48 views

Solution to Cauchy Problem

I am trying to solve the following Cauchy Problem: $y'(t) = A(t)y(t), A=\begin{pmatrix} t &-1 \\ 1 &t \end{pmatrix}, y(0)=y_0$ What I did: I know that $ \forall\ t, s \in\ \mathbb{R}:...
2
votes
1answer
53 views

Exisence and unicity for Cauchy Problem

i have the following theorem: Let the Cauchy problem $$ y'=f(x,y),~~ y(x_0)= y_0 $$ in $$ R=\{(x,y) \in \mathbb{R}^2: |x-x_0| \leq a, |y-y_0| \leq b\} $$ If $f$ is continuous and bounded ...
2
votes
1answer
162 views

Deducing the existence of a PDE by constructing it inductively via its Taylor series expansion

Consider the differential equation $$u_{t t} = c^2 u_{x x} - k u, \qquad k, u > 0$$ together with the Cauchy data $$u(x, 0) = e^x, u_t(x, 0) = 0$$ I now want to find a solution $u: \mathbb R^...
2
votes
0answers
20 views

Where is the solution uniquely determined by the data?

Question: Solve the PDE $$y\frac{\partial u^2}{\partial x^2} + (y-x)\frac{\partial u^2}{\partial x \partial y} -x \frac{\partial u^2}{\partial y^2} = \frac{y-x}{y+x}\bigg(\frac{\partial u}{\partial ...
2
votes
0answers
47 views

Uniqueness of solution for linear first-order partial differential equations

In Polyanin's Handbook of First Order Partial Differential Equations (2002), in Section 10.1.2, it is stated that the non-homogeneous linear, first-order partial differential equation: $$\sum_{i=1}^...
2
votes
1answer
153 views

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 ...
2
votes
0answers
34 views

PDE Cauchy Problem w/ Two Boundaries

I'm trying to solve this Cauchy problem, which appears to be rather basic, but it seems like I'm missing something. The problem is as follows: $ u_x+u_t+u=0 \\ 0<\alpha*t<x<\infty \\ \...
2
votes
1answer
52 views

Solve $\dot X = AX$ and find the time at which the area doubles.

We have a linear operator $X(t): \mathbb{E}^2 \to \mathbb{E}^2$ such that $$\dot X = AX\quad X(0)=\mathbf{1}$$ To be clear $X$ is a $2\times 2$-matrix. An ink spot is contained in $\mathbb{E}^2$ at $t=...
2
votes
1answer
73 views

Reaction-diffusion Cauchy problem

How do I solve this reaction-diffusion Cauchy problem? $$\left\{ \begin{array}{l l} u_{t} - \kappa u_{xx} +ru=0 & \quad \mbox{$x \in \mathbb{R}, t>0$,}\\ \quad u(x,0) = \phi(x), \end{array} \...
2
votes
0answers
100 views

Computation of a wave equation using Kirchoff's formula !!

I want to solve this wave equation in $3$-D using Kirchoff's formula but all I've done it doesn't work. In particular I'm not able to compute the integral in Kirchoff's formula. you can see the ...
2
votes
1answer
41 views

Explaining results involving differential equations using the theorem for the existence and uniqueness of the solutions of a Cauchy Problem

Let $f(x,y) = 3(y-1)^{2/3y}$. I want to show that $y' = f(x,y)$ has two solutions for $y_0 = y(0) = 1$ and one solution for $y_0 = y(0) = 2$. I am trying to understand the solution to this problem. ...
2
votes
2answers
382 views

Cauchy-Riemann equations, $f(z) = \bar{z} \sin z$

Find all $z \in \mathbb{C}$ such that the function $f(z) = {\bar z}\sin(z)$ satisfies the Cauchy-Riemann equations. Own work: Let $z = x+iy$ where $x$ and $y$ are real numbers Thus, ${\bar z} = x-...
2
votes
0answers
307 views

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 ...
2
votes
1answer
94 views

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 ...
1
vote
1answer
53 views

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 (...
1
vote
2answers
72 views

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; ...
1
vote
2answers
40 views

Solve the Cauchy problem for the linear PDE

I have the linear PDE $$yu_x - xu_y = 0 \qquad x^2 + y^2 < a^2 \\ u(0,y) = (a^2-y^2)^{\frac{1}{2}} \qquad y \in(-a,a)$$ where $a > 0$ is a constant. So what I have done is to say that $$a_1 ...
1
vote
1answer
103 views

Cauchy problem with real parameter?

I guys. I'm doing this problem: $$\begin{cases} y'(x)=\frac{2y(x)}{x}+3x^{\alpha} \\ y(1)=2\\ \end{cases} $$ with $\alpha\in \mathbb R^{+}$ constant. I have started studying the existence and ...
1
vote
1answer
3k views

Find limit by using Cauchy's second limit theorem

I have to find the limit of this problem. I have found that with problems like this, I always get wrong answers when I try to use Cauchy's second limit theorem. I do not understand how to solve this ...
1
vote
2answers
112 views

If $\vert F(t,x)\vert \leq \alpha (t)\vert x\vert + \beta(t)$ the maximal solutions are global for an ODE

Let $F: \mathbb{R} \times\mathbb{R}^2 \rightarrow \mathbb{R}$ localy lipschitz in its second variable. Let the Cauchy problem be: $$x'=F(t,x),\\ x(0) = x_0 $$ Let $\alpha : \mathbb{R} \rightarrow \...
1
vote
2answers
53 views

Solve for $u$ the PDE $(x − y − 1)u_x + (y − x − u + 1)u_y = u$ if $u=1$ on $x^2+(y+1)^2=1.$

Solve the Cauchy problem $$(x − y − 1)u_x + (y − x − u + 1)u_y = u,$$ if $u=1$ on $x^2+(y+1)^2=1.$ Attempt. $$\frac{dx}{x-y-1}=\frac{dy}{y-x-z+1}=\frac{dz}{z}$$ so $$\frac{dx+dy}{(x-y-1)+(y-x-z+1)}...
1
vote
1answer
54 views

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 ...
1
vote
2answers
99 views

for the cauchy problem , determine unique solution ,or no solution , or infinitely many solution

for the cauchy problem , determine unique solution ,or no solution , or infinitely many solution $u_x-6u_y=y$ with the date $u(x,y)=e^x$ on the line $y=-6x+2$ My attempt: given $u_x-6u_y=y$ then $...
1
vote
1answer
165 views

Existence and uniqueness of the solution for the Cauchy problem for ODE system.

Please help me to solve the following problem: I got the system of differential equation $$\dot x_1=2t \sqrt{|x_1|}+\frac{1}{\sqrt{x_2-1}}, \dot x_2=x_1x_2.$$ I also got initial conditions: $(t_0,...
1
vote
1answer
89 views

Laplace transform of a Cauchy's problem

I have to determine the Laplace transform of $u$, where $u$ is the solution to this Cauchy's problem $$ u'(t) + \int_0^t e^{t-s} \left( \int_0^s u(r) ~ dr \right) ~ ds = 0, t > 0, u(0) = 1$$ $L[u](...
1
vote
3answers
317 views

How to find matrix exponential $e^A$

I have the matrix $$A =\begin{pmatrix} 0 & 1 \\ - 1 & 0 \end{pmatrix}$$ and I have to find $e^A$ I've found two complex-conjugate eigenvalues $\lambda_{1,2} = \pm i$ so substracting $\...
1
vote
1answer
39 views

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: $$y(x) = C_1e^{-x}+C_2e^{3x}+\frac{1}{5}e^{4x}$$ ...
1
vote
1answer
124 views

Unique solution of Cauchy problem in a neighbourhood of given set

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 $...