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Questions tagged [singular-solution]

For questions about the singular solution of the ordinary differential equations. It is a special type of solution different from general solution. Such solutions does not contain any arbitrary constant and is not a particular case of the general solution.

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

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)=\...
Renato lorentz's user avatar
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1 answer
64 views

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

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 ...
X.H. Yue's user avatar
2 votes
0 answers
115 views

How to get a Filippov solution?

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)=...
Liu C's user avatar
  • 21
1 vote
1 answer
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Obtaining correct boundary conditions around a pole for Bessel equation plus delta function

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 ...
Andreas Christophilopoulos's user avatar
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0 answers
41 views

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

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 \...
jozinho22's user avatar
  • 127
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0 answers
275 views

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

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 ...
Thomas Finley's user avatar
12 votes
5 answers
652 views

Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$

Closed-form solutions to $x''+\frac{k}{m}\ x+\mu\ g\ \text{sgn}(x')=0$ Introduction______________________ I am looking for simple mechanics models that could have closed-form solutions that achieves ...
Joako's user avatar
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1 vote
1 answer
32 views

What is the general solution ( without start terms ) of laplace equation - $u_{ss}=-u_{tt}$

I had a question is Analysis fourier, which was: let $u(x,y)$ such that $u_{xx}-2u_{xy}+5u_{yy}=0$ when $s=x+y$ and $t=2x$ I solved it, and reached the conclusion of the Laplace equation ( it is true, ...
LearningToCode's user avatar
1 vote
1 answer
52 views

How to find singular solutions from a general one

Find singular solutions, if given general solution \begin{equation*} y= Ce^x + \dfrac{4}{C} \end{equation*} Usually after solving the equation you will use the parameter for finding the singular ...
Maroon Racoon's user avatar
3 votes
1 answer
215 views

Proving that these solutions are formally solving these differential equations: $x'' = -\text{sgn}(x')$ and $y'' = \sqrt{|y'|}$

Please take a look also to the comments section, here, and in other people answers, since there are extended what are my apprehensions about the validity of the found answers. I have found these two ...
Joako's user avatar
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0 votes
0 answers
160 views

Prove only 1 solution exists for the following equation $x^2 = y^2 + 71$ where $x, y$ and pos integers

Hi I've been stuck on this equation for a while. Pretty sure it has something to do with Pell's equation $x^2 - D y^2 = a$. Not exactly sure. The question: (i)Prove that there is only one solution to $...
Alex's user avatar
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0 answers
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quasi-solvable ode

I face with a quasi-solvable ode $x^{2}(1+x^{2})y''+2xy'+A(1+x^{2})^{2}y=0$ where $A$ is a constant. I am trying to find a solution for this ode. My suggestion is that we can rewrite the above ...
Mehdi's user avatar
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23 votes
2 answers
838 views

Are these equations "properly" defined differential equations? (finite duration solutions to diff. eqs.)

Are these equations properly defined differential equations? Modifications were made to a deleted question to re-focus it. I am trying to find out if there exists any exact/accurate/non-approximated ...
Joako's user avatar
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2 votes
2 answers
734 views

Singular solution of differential equation $p^2y+2px-y=0$

I have the following differential equation before me for which I am required to find singular solution: $p^2y+2px-y=0$ where $p=\dfrac{dy}{dx}$ On solving it, I got the following general solution: $y^...
HARVEER RAWAT's user avatar
2 votes
1 answer
197 views

How to find the singular solution of this DE?

How to find the singular solution of $8ap^3=27y$ where $p=\frac{dy}{dx}$? My effort: $8ap^3=27y$ Differentiating both sides we get $8a(3p^2)\frac{dp}{dx}=27p \implies 8ap^2\frac{dp}{dx}=9p\implies p(...
Olivia's user avatar
  • 841
0 votes
1 answer
96 views

Frobenius method solution

Suppose I have a second order linear differential equation. I have a solution about a regular singular point say $x=0$. Suppose the indicial equation has a repeated root for the indicial equation. I ...
Upstart's user avatar
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0 votes
1 answer
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Is $r(t) = \frac{1}{144}(T-t)^4\theta(T-t)$ a valid solution to $\ddot{r}=\sqrt{r},\,r(0)=\frac{T^4}{144}>0$? (with $\theta(t)$ the Heaviside step fn)

Is $r(t) = \frac{1}{144}(T-t)^4\theta(T-t)$ a valid solution to $\ddot{r}=\sqrt{r},\,r(0)=\frac{T^4}{144}>0$? (with $\theta(t)$ the Heaviside unitary step function) I am looking here for examples ...
Joako's user avatar
  • 1,424
1 vote
1 answer
134 views

Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$?

Does $x(t) = \exp\left(\frac{t}{t-1}\right)\cdot\theta(1-t)$ solve $\dot{x}\cdot(1-t)^2+x=0,\,\,x(0)=1$ with $\theta(t)$ the standard unitary step/Heaviside function $$\theta(t) := \begin{cases} 0 &...
Joako's user avatar
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1 vote
1 answer
145 views

How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$?

How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? (with $\theta(t)$ the standard unitary step function). I have found the ...
Joako's user avatar
  • 1,424
6 votes
0 answers
203 views

Is the solution to $\theta''+0.021\,\text{sgn}(\theta')\sqrt{|\theta'|}+0.02\sin(\theta)=0,\,\theta_0=\pi/2,\,\theta'_0=0$ of finite duration?

Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration? I ...
Joako's user avatar
  • 1,424
1 vote
1 answer
212 views

On the existence of singular solution for an ODE

I feel some thing unclear in finding the singular solution ($y=0$, by $p$-discriminant method) for the following differential equation $$f \left(x,y,y'\right)=xy'+2y=0,$$ while the general solution ...
Riaz's user avatar
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0 votes
3 answers
136 views

Need analytical solution to this ODE

I have two ordinary differential equations that I'm trying to solve analytically, if a solution exist. I'm not interested in numerical solutions. Here they are: \begin{align} \frac{\ddot{a}}{a} - \...
Cham's user avatar
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1 vote
2 answers
271 views

Find the singular points of the ODE

Consider the ODE $$\frac{d^2x}{dt^2}+e^t\frac{dx}{dt}+\frac{x}{1+2t}=0$$ What are the singular points of this ODE? For an ODE of the form $$P(t)\frac{d^2x}{dt^2}+Q(t)\frac{dx}{dt}+R(t)x=0$$ a point is ...
James Anderson's user avatar
0 votes
0 answers
53 views

Find the general solution of the second ordinary linear homogeneous differential equation when a singular solution is a second order polynomial.

I am having problem solving an ODE. This is the equation: $(x-1)y'' - (x+1)y' + 2y = 0$ So I started with y1 = ax^2 + bx + c, and since this is a solution I substituted its first and second order ...
Ana's user avatar
  • 1
1 vote
1 answer
140 views

Help Finding Explicit Solution Non Linear Differential Equation

I am wondering if anyone has any advice to try to find an explicit solution (or some approximation) for the following non-linear differential equation system: $\frac{dx}{dt} = A - B y(t)$ $\frac{dy}{...
etimi's user avatar
  • 11
2 votes
1 answer
289 views

Types of singularities in ODE

Consider the nonlinear ODE $$y\cdot\frac{d^{2}y}{dx^{2}}-\left(\frac{dy}{dx}\right)^{2}+1=0$$ with $y : \mathbb{R} \to \mathbb{R}$. As seen, it does not satisfy Picard–Lindelöf theorem because of ...
Maksim Surov's user avatar
4 votes
0 answers
306 views

It is possible to find a solution to $y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0,$ $\,y'(0)=0,\,y(0)= 1/4$?

It is possible to find an exact solution (hopefully in "close form") to $$y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4$$? How?... There ...
Joako's user avatar
  • 1,424
-1 votes
2 answers
429 views

The differential equation $y'=y+1$ has no singular solution?

$\frac{dy}{dx}=y+1$ Solving the above given differential equation, yields the following general solution. $y+1=e^{x+C}$ $y=Ce^{x}-1$ $\implies$ Solution $y=-1$ at $C=0$ Can I say that $y= -1$ is a ...
marceux's user avatar
11 votes
1 answer
249 views

Understanding the singularity in $f^{-1}(x)=\int_0^x f(t)dt$

Note that here $f^{-1}(x)$ is the functional inverse of $f$. Clearly from the definition of the equation $f^{-1}(0)=0\Rightarrow f(0)=0$. By setting $x\to f(x)$ and differentiating one finds $$f'(x)f(...
K. Grammatikos's user avatar
2 votes
2 answers
268 views

Finding the exact solutions for a system of equations

I am computationally trying to solve the system of linear equations given below. $ \frac{1}{1+tan(\theta/2)*tan(\phi/2)*tan(\psi/2)} \begin{bmatrix} tan(\psi/2)-tan(\theta/2)*tan(\phi/2) \\ tan(\phi/...
Jules's user avatar
  • 27
5 votes
0 answers
242 views

General form of the solutions to a PDE

I have encountered the following linear PDE in the context of reaction-diffusion processes \begin{equation} \partial_t (\nabla^2f) - \nabla\cdot \left( \nabla (\nabla^2 f) - m^2 \, \nabla f \right) =...
Saïd M's user avatar
  • 416
0 votes
0 answers
66 views

prove unique solution for nonhomogeneous PDE

I'm trying to prove a unique solution for this PDE \begin{align} u_{tt} (x,t)+(d+c) u_{xt} (x,t)+cdu_{xx} (x,t) &= F(x,t), \quad x \in \mathbb{R}, t > 0 \\ u(x,0) &= f(x) \\ u_t(x,0) &= ...
David's user avatar
  • 161
1 vote
0 answers
59 views

Singular solution

I am trying to solve this differential equation, I have already obtained the general solution of the given equation in parametric form, however I do not know where to get the particular solution y = -...
user895583's user avatar
0 votes
2 answers
673 views

How to determine if this Differential Equation has a singular solution?

I found this Differential Equation $y'^2=16x^2$ and the text says it has a singular solution... first they make an implicity derivation with respect to $y'$ then they have $2y'=0$. Therefore $y=...
user760637's user avatar
1 vote
1 answer
312 views

Singular solution of the differential equation $(y')^2-3xy'+y^2=0$.

To find singular solution of the differential equation $$(y')^2-3xy'+y^2=0$$ I find its $p $-discriminant as $$y=\pm \frac {3}{2}x $$ but none of the factor satisfies differential equation. How to ...
Ymylife's user avatar
  • 191
2 votes
1 answer
97 views

Studying singular solutions of the ODE

Given that $$8x(y')^3 + 12y(y')^2 - 9y^5 = 0$$ Find all solutions and study singular solutions. My solution After noting that $y' = 0$ is solution only when $y = 0$ we subtitute $y' = \frac{1}{x'}$ ...
IPPK's user avatar
  • 205
2 votes
1 answer
457 views

Relation between general solutions and singular solution of Clairaut’s equation.

So I'm trying to do this proof, The form of Clairauts equation is $$y(x) = xy' + f(y')$$ You differentiate once to get $$ y' = y' + xy'' + f'(y')y''$$ You rearrange and get two solutions The general ...
Cuharious's user avatar
2 votes
1 answer
87 views

Solving an ODE with singular point: is this correct?

This is from a text book by Tsai Tai-Peng "Lectures on Navier-Stokes Equations", page 148 if anyone wants to read. But I am only asking about how to solve this ODE, $$(1-t^2) L'' +2L+LL' = 0, \quad t\...
Calvin Khor's user avatar
  • 35.1k
-3 votes
1 answer
56 views

How to solve : $(3x+5)\Big(\frac{\mathrm{d}y}{\mathrm{d}x}\Big)^2-(3y+x)\Big(\frac{\mathrm{d}y}{\mathrm{d}x}\Big)+y=0$? [closed]

How to solve this ordinary differential equation? $$ (3x+5)\Big(\frac{\mathrm{d}y}{\mathrm{d}x}\Big)^2-(3y+x)\Big(\frac{\mathrm{d}y}{\mathrm{d}x}\Big)+y=0 $$ How to find the general solution and ...
serenity's user avatar
1 vote
1 answer
128 views

How to show singular Sturm-Liouville problem has no eigenvalues?

I have the following SL problem: $$ (x^{2} f')'+ \lambda f = 0 $$ where $ \lvert f(x) \rvert $ is bounded as $ x \rightarrow 0 $ and $ f(1) = 0 $ I have to show that the above problem has NO ...
Uncle Ben's user avatar
0 votes
1 answer
244 views

Unique Solution Periodic Differential Equation

I am reading in a book and I encountered this step that I do not understand in the proof. The book is proving the following: $ \frac{dx}{dt} = A(t) x + f(t) $ where $A(t)$ is a continuous $n \times ...
Jason Chiu's user avatar
0 votes
1 answer
271 views

Singular solution of first order differential equation [closed]

The singular solution of $y = px + p^3,$ where, $ p = dy/dx$. I have the answer $4x^3 + 27y^2 = 0.$ as given in the key, but I am not getting this answer when using $p$- discriminant method.
sabeelmsk's user avatar
  • 612
0 votes
1 answer
123 views

Solving differential equation $2xy + 2x +(x^2-1)y'=0$ and singular solutions

The solution of separable differential equation $2xy + 2x +(x^2-1)y'=0$ is $y = \frac{c+ x^{2}}{1- x^{2}} $, $c$ is a constant. When separating variables we have to divide by $(1-x^{2})(y+1) $. ...
user121's user avatar
  • 527
1 vote
1 answer
72 views

How to find singular solutions and determine their types for system of equations.

Given equations $x^\prime=x(1-x-2y)$ and $y^\prime=y(1-y-4x)$ find all the singular solutions in the upper quadrant, $x\geq 0, y\geq 0$ and determine the type and stability. So for singular solutions ...
AColoredReptile's user avatar
3 votes
1 answer
532 views

Proof that Singular Solution of Clairaut's Equation is the envelope of the family of General Solutions [duplicate]

I want to show that the singular solution is the envelope for the general solutions. Proof Outline Both solutions pass from the same point $(a,b)$ Both solutions have the same gradient at that ...
DustyLynx's user avatar
4 votes
2 answers
235 views

Behavior of a function near a singular point

What is the correct wording concerning the behavior of a function near a singular point? For example, the function $$f(x)=\frac{e^x+2}{x^2}$$ behaves like $2/x^2$ as $x$ approaches zero. Often, one is ...
Gateau au fromage's user avatar
-2 votes
1 answer
130 views

How do you find singular solutions to first order differential equations?

How do you find singular solutions to differential equations? for example I am working with the equation $$\frac{dy}{dx}=y\cos^2(x)$$ This is a very easy separable equation. For some reason by ...
K. Gibson's user avatar
  • 2,391
6 votes
1 answer
178 views

Euler (equidimensional) equation question

Consider the equation $$x^2y''-8xy'+20y=0.$$ From an undergraduate ODE course, it is known that the two linearly are $y_1=x^5$ and $y_2=x^4$. However, why don't we consider solutions, for example, ...
Gateau au fromage's user avatar
1 vote
0 answers
1k views

Why don't linear differential equations have any singular solutions?

A function $\phi(x)$ is called the singular solution of the differential equation $F(x,y,y')=0,$ if uniqueness of solution is violated at each point of the domain of the equation. Geometrically this ...
nmasanta's user avatar
  • 9,302
3 votes
2 answers
5k views

Singular solution of $y^2(y - xp) = x^4p^2$

Given differential equation, $y^2(y - xp) = x^4p^2$ where {$p = dy/dx$} To find the singular solution, I have extracted the p-discriminant relation which is, $y^3x^2(y + 4x^2) = 0$ From here it is ...
Hari Prasad's user avatar