Questions tagged [singular-solution]

Question 1

Motivation: I am trying to find and plot the Geodesics on the Torus, and obtained the following ODE: $$ \dot{\theta}(\varphi)=f_P(\theta) \qquad f_P(\theta)=\frac{1}{Pr}(R+r\cos\theta)\sqrt{(R+r\cos\...

Question 2

On page 48 of [Zill2022], there is an initial-value problem (IVP): $$ \frac{dy}{dx}=xy^{1/2}, y(0)=0. \tag{1} $$ I try to solve this IVP by myself. My solution is as follows: Solution: First, it is ...

Question 3

Exercise Question given in text book : For the question (ii) here, from the equation $(2)$, we could eliminate $a$ and $b$ and say that $xy=1$ is the singular solution. But that was not done here. As ...

Question 4

I have the following 'nonlinear heat equation' where I am trying to compare the exact solution with the answer which one would get using lowest-order perturbation theory: $$ {\frac {\partial }{\...

Question 5

I'm trying to solve the following partial differential equation: $$z= 3x \frac{\partial z}{\partial x} + 3y \frac{\partial z}{\partial y} + \frac{1}{\frac{\partial z}{\partial x} \frac{\partial z}{\...

Question 6

Consider the following recurrence relation $$-2A_{n-1} + (n-1)A_{n+1} + (2-n^2) A_{n} =0$$ Notice that in this recurrence relation there is a singularity at $n=1$. I was wondering if these recurrence ...

Question 7

Suppose I have a real symmetric $M\times M$ matrix whose matrix elements are parameterized by linear functions $$A_{ij}(x_1,\cdots,x_N) = \sum_{n=1}^N c_n^{ij} x_n.$$ For fixed real coefficients $c_n^{...

Question 8

I am trying to solve the following ODE, where s is the independent variable, and k is a parameter $$ \left( {\frac {{\rm d}^{2}}{{\rm d}{s}^{2}}}T \left( s \right) \right) {s}^{3}+ \left( -2\,k{s}^{...

Question 9

I am learning the p-discriminant method for finding singular solutions for non-linear first order ODE's. But in this example ode, which is d'Alembert ODE, when using standard d'Alembert method for ...

Question 10

Let be $A\in R^{n \times n},x_0,c,d\in R^n$ and $\alpha \in R-\{0\}$. Prove that the system $$x'=Ax+\cos(\alpha t)c+\sin(\alpha t)d $$ $$x(0)=x_0$$ has a particular solution of the form $\varphi_p(t)=\...

Question 11

Kindly help me in the following: Let $C_0=A \times B$ and $C_1=B \times A$, where $A$ is a full rank square matrix, $B$ is a non-zero singular square matrix. Entries in $A,B,C$ are from finite ...

Question 12

Recently, I read a book ISSN 2195-9862 about the Filippov theory. There is a differential inclusion $$\dot{x}\in F(x)=\begin{cases} -1&x>0\\ [-1,1]&x=0\\ 1&x<0 \end{cases}\\ x(t_0)=...

Question 13

Consider the following Bessel-like equation: $$\frac{\partial^2u}{\partial \rho^2}+\frac{1}{\rho}\frac{\partial u}{\partial \rho}+u=\frac{A}{\rho}\delta(\rho) $$ Here, $\rho$ is non negative. This ...

Question 14

On the theme of differential equations, I wonder why we still need to keep the solution of the homogeneous equation. For example the linear differential equation : $y' + ay = x^2 \enspace \enspace \...

Question 15

Find the singular solution of the differential equation $$4xp^2=(3x-1)^2,$$ where $p=\frac{dy}{dx}.$ As we know the singular solution, of a first order differential equation, is represented by the ...

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Questions tagged [singular-solution]

Is there a way to rewrite a differential equation to obtain uniqueness? Lipschitz

How to solve $\frac{dy}{dx}=xy^{1/2}, y(0)=0$?

Is it a must for a singular solution to have the dependent variable in it in case of PDE?

Comparison of exact and perturbed limits of solving a PDE

How to solve this Clairaut-type PDE: $z = 3x p + 3y q + \frac{1}{p q z^6}$?

Recurrence relations with singularities

Intersections of degenerate eigenvalue surfaces for a parametrized symmetric real matrix

ODE with additional parameter - Comparison of various limiting solutions

Why p-discriminant method for finding singular solution does not produce the same singular solution as d'Alembert method?

Particular solution $x'=Ax+\cos(\alpha t)d+\sin(\alpha t)d $

How to find the matrix $A$ from $C_0=A \times B$ or $C_1=B \times A$, given the singluar matrix B.

How to get a Filippov solution?

Obtaining correct boundary conditions around a pole for Bessel equation plus delta function

Why do we need to keep the solution of the homogeneous equation in the general equation?

Find the singular solution of the differential equation $4x(\frac{dy}{dx})^2=(3x-1)^2.$

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