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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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 \...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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^...
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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(...
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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 ...
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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 ...
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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
$$\theta(t) := \begin{cases} 0 &...
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How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$?
How to "formally" prove that $x(t)=\frac{1}{4}(2-t)^2\theta(2-t)$ solves $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$? (with $\theta(t)$ the standard unitary step function).
I have found the ...
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Is the solution to $\theta''+0.021\,\text{sgn}(\theta')\sqrt{|\theta'|}+0.02\sin(\theta)=0,\,\theta_0=\pi/2,\,\theta'_0=0$ of finite duration?
Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration?
I ...
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On the existence of singular solution for an ODE
I feel some thing unclear in finding the singular solution ($y=0$, by $p$-discriminant method) for the following differential equation
$$f \left(x,y,y'\right)=xy'+2y=0,$$
while the general solution ...
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Need analytical solution to this ODE
I have two ordinary differential equations that I'm trying to solve analytically, if a solution exist. I'm not interested in numerical solutions. Here they are:
\begin{align}
\frac{\ddot{a}}{a} - \...
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2
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Find the singular points of the ODE
Consider the ODE
$$\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$$
a point is ...
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Find the general solution of the second ordinary linear homogeneous differential equation when a singular solution is a second order polynomial.
I am having problem solving an ODE.
This is the equation:
$(x-1)y'' - (x+1)y' + 2y = 0$
So I started with y1 = ax^2 + bx + c, and since this is a solution I substituted its first and second order ...
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Help Finding Explicit Solution Non Linear Differential Equation
I am wondering if anyone has any advice to try to find an explicit solution (or some approximation) for the following non-linear differential equation system:
$\frac{dx}{dt} = A - B y(t)$
$\frac{dy}{...
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Types of singularities in ODE
Consider the nonlinear ODE $$y\cdot\frac{d^{2}y}{dx^{2}}-\left(\frac{dy}{dx}\right)^{2}+1=0$$ with $y : \mathbb{R} \to \mathbb{R}$.
As seen, it does not satisfy Picard–Lindelöf theorem because of ...
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Singular solution of second order partial differential equations
I know that complementary function, the particular integral can be obtained of a second-order partial differential equation.
I know that a singular solution to first order pde can be obtained.
Is ...
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It is possible to find a solution to $y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0,$ $\,y'(0)=0,\,y(0)= 1/4$?
It is possible to find an exact solution (hopefully in "close form") to $$y''+\sqrt{|y|}\operatorname{sgn}(y)+\sqrt{|y'|}\operatorname{sgn}(y')=0, \,y'(0)=0,\,y(0)= 1/4$$? How?...
There ...
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The differential equation $y'=y+1$ has no singular solution?
$\frac{dy}{dx}=y+1$
Solving the above given differential equation, yields the following general solution.
$y+1=e^{x+C}$
$y=Ce^{x}-1$
$\implies$ Solution $y=-1$ at $C=0$
Can I say that $y= -1$ is a ...
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Understanding the singularity in $f^{-1}(x)=\int_0^x f(t)dt$
Note that here $f^{-1}(x)$ is the functional inverse of $f$. Clearly from the definition of the equation $f^{-1}(0)=0\Rightarrow f(0)=0$. By setting $x\to f(x)$ and differentiating one finds
$$f'(x)f(...
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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/...
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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) =...
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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) &= ...
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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 = -...
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2
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502
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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=...
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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 ...
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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 $y' = \frac{1}{x'}$ ...
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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 ...
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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,
$$(1-t^2) L'' +2L+LL' = 0, \quad t\...
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How to solve : $(3x+5)\Big(\frac{\mathrm{d}y}{\mathrm{d}x}\Big)^2-(3y+x)\Big(\frac{\mathrm{d}y}{\mathrm{d}x}\Big)+y=0$? [closed]
How to solve this ordinary differential equation?
$$
(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
$$
How to find the general solution and ...
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How to show singular Sturm-Liouville problem has no eigenvalues?
I have the following SL problem:
$$ (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 ...
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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 ...
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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.
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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) $. ...
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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 ...
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1
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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 ...
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2
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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}$$ behaves like $2/x^2$ as $x$ approaches zero. Often, one is ...
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1
answer
125
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How do you find singular solutions to first order differential equations?
How do you find singular solutions to differential equations? for example I am working with the equation $$\frac{dy}{dx}=y\cos^2(x)$$ This is a very easy separable equation. For some reason by ...
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1
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Euler (equidimensional) equation question
Consider the equation $$x^2y''-8xy'+20y=0.$$ From an undergraduate ODE course, it is known that the two linearly are $y_1=x^5$ and $y_2=x^4$. However, why don't we consider solutions, for example, ...
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0
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Why don't linear differential equations have any singular solutions?
A function $\phi(x)$ is called the singular solution of the differential equation $F(x,y,y')=0,$ if uniqueness of solution is violated at each point of the domain of the equation. Geometrically this ...
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2
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Singular solution of $y^2(y - xp) = x^4p^2$
Given differential equation, $y^2(y - xp) = x^4p^2$ where {$p = dy/dx$}
To find the singular solution,
I have extracted the p-discriminant relation which is, $y^3x^2(y + 4x^2) = 0$
From here it is ...
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1
answer
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Significance of the term "general" in a general solution of the ODE.
General Solution: A solution of a differential equation in which the number of arbitrary constants is equal to the order of the differential equation is called the general solution or complete ...
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1
answer
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Can a first order homogeneous ODE have singular solution(s) (free of a constant)
Can $$y'=\frac{px+qy}{rx+sy}$$ have singular (essential) solution(s) free of a constant of integration? Here, $p,q,r,s$ are independent of $x$ and $y$.
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Clairaut's equation: Find the general solution of $x^2 (y-xy') =y(y') ^2$ if the singular solution doesn't exist.
Question:
Find the general solution of
$$x^2 (y-xy') =y(y') ^2$$
if the singular solution doesn't exist.
Now, I know that it has to be solved by Clairaut's equation.
However, the given equation ...
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