Questions on (ordinary) differential equations. For questions specifically concerning partial differential equations, use the (pde) tag.

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2
votes
2answers
23 views

General form for finding tangent that intersects a point not on the curve

Particular cases of this problem have previously been addressed here and here, but I'm interested in the general case of the following problem: Given a function $f(x)$ and a point $P = (x_0, ...
1
vote
1answer
31 views

Statement about solutions for diff. equations.

Let $f_1,f_2:\mathbb{R}^2\to\mathbb{R}$ be $C^{\infty}$ functions such that $f_1(x,y)\leq f_2(x,y)\; \forall(x,y)\in\mathbb{R}^2$. Suppose that $\psi_1:I_1\to \mathbb{R}$ and $\psi_2:I_2\to ...
0
votes
1answer
35 views

Differential equation $x'=11x -x^2 -24$

Getting stuck finding $x(t)$ on the differential equation: $dx/dt = 11x -x^2 -24$ with $x(0)=5$. So my work so far is: $dx/(11x-x^2-24) = dt$ Using partial fractions $A(x-3) + B(x-8) = 1$, so $A = ...
0
votes
1answer
25 views

Differential equation $xy'+2y=0$ and the form of arbitrary constant in its general solution

If I'm solving the differential equation in the title I will get to: $$\log(y)=-2\log(x)+c$$ then I'll get $y=e^c/x^2$ eith arbitrary constant $c$. So I know I can write $y=d/x^2$ where $d$ is an ...
0
votes
1answer
17 views

Prove that Autonomous are invariant under time translation

Reading my way through a big boy ODE book, and the authors write It is clear that if $\varphi(t)$ is a solution to $x'=f(x) \quad x(t_0)=x_0$, then clearly $\varphi(t+t_0)$ is a solution to $x'=f(x) ...
2
votes
1answer
38 views

How to prove a tempered distribution is in $L^p(\mathbb{R}^n)$

Given $g \in L^p(\mathbb{R}^n)$, how can I to prove that the tempered distribution $$f=\mathcal{F}^{-1}[(z-4\pi^2|x|^2)^{-1}\mathcal{F}g]$$ is in $L^{p}(\mathbb{R}^n)$ where $z \in \{u \in ...
7
votes
1answer
181 views

Properties of the solutions to $x'=t-x^2$

Let $f_c$ be the solution to $$ \left\{ \begin{array}{c} x'=t-x^2 \\ x(0) =c \end{array} \right. $$ I'm trying to prove: If $c \geq 0$ then $f_c(t)$ is defined for all $t>0$ There is a ...
1
vote
1answer
35 views

Heat equation: Why are these ratios of functions constant

One can solve the heat equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} $$ by a separation of variables such that $u(x, t) = f(x)g(t)$. Substituting this into the ...
0
votes
2answers
33 views

If $z'\le az+b$ then $z(t)\le z_0+bt$

If $z$ satisfies; $z'\le az+b$, $\ z(0)=z_0>0$ with constants $a,b$ why is true that $z(t)\le z_0+bt$, if $a=0$ It is clear that it can't be justified only by integrating. We had only Gronwall ...
0
votes
2answers
42 views

How to solve DE $y'=1/(x^2y)(y^2-1)^{3/2}$

Man, I'm having troubles with this differential equation. I just can't do any math if I'm tired... What I have done: $$\frac{y'\cdot y}{(y^2-1)^{3/2}}=\frac{1}{x^2}$$ Now I integrated both sides from ...
4
votes
2answers
215 views

On why we have $dy = f'(x)dx$?

I am following Ordinary differential equations by Tenenbaum. Page 48 The differential is defined as: $$dy(x,\Delta x) = f'(x) \Delta x$$ Note: we will want to apply this definition to the function ...
0
votes
0answers
28 views

Rerformulation of a previous question concerning a problem in physics that involves integration of 2-forms over the sphere

In this question the integral proposed in the posting concerns a physical problem that can shortly be described by the following : Let $J$ be a real valued function on the sphere (in fact it is a ...
-1
votes
3answers
72 views

Is $f(x)=x$ the solution of an integral equation? [closed]

Suppose that $f:[0, \infty)\longrightarrow \mathbb{R}$ is continuous and $f(x) \neq 0 $ for all $x>0$. If $$ \big(\,f(x)\big)^2=2 \int_0^x f(t)\,dt, $$ for all $x>0$, is it then true that ...
0
votes
1answer
96 views

Parameterise $y^2-x^2=1$ - not possible.

I'm doing stuff from a book and it has just spoke of the importance of not parameterising half a curve (with the example of a circle). However I am not sure what to do. First of all ...
1
vote
0answers
43 views

Differential equation with $x'(t)=\sqrt[5]{(x(t))}$

Solve the following Cauchy problem for $t\in\mathbb R$ $x'(t)=\sqrt[5]{(x(t))}$ $x(0)=0$ Is the solution unique ? Now this is a differential equation of the form $x'=f(x)$, thus; ...
0
votes
0answers
14 views

why isn't this part of the equation considered transient?

In the expression $y=\frac{1}{2}x^{-2}e^x+cx^{-2}e^{-x}$ the answer key says only $cx^{-2}e^{-x}$ is transient. Why wouldn't $\frac{1}{2}x^{-2}e^x$ be transient, since $x^{-2}=\frac{1}{x^2}$ which ...
0
votes
1answer
13 views

Separable differential equation: Particular solutions given initial conditions.

If given initial conditions for a separable differential equation, finding them is easy enough, for example if $y(0) = 0$ But how do I know that these are the only solutions? Or that this solution ...
0
votes
0answers
17 views

Moment matching, differential equation

My task is to apply moment matching method on the wave equation $Mu''+Cu'+Ku=0$. When I read materials about moment matching it's always considered the system $x'=Ax+Bu, y=Cx$. I don't understand ...
3
votes
0answers
47 views

How to solve this complicated ordinary differential equation?

Consider the following non-linear ODE: $$x^2 \frac{dy}{dx} + \exp{\left(x \, \frac{d^2 y}{dx^2} \right)} = \sin \left(\frac{d^3y}{dx^3} \, \cos \left( \frac{d^4y}{dx^4} \right) \right) $$ I have no ...
2
votes
3answers
96 views

Is the function $y(t)$ is a solution of the equation $y'=\sin(yt)$?

Is the function $y(t)$ a solution of the equation $y'=\sin(yt)$? any thought to start me up? I'm not sure what is the question asking. EDIT: Someone tell me if I'm correct or not . If I'm finding ...
6
votes
1answer
160 views

Finite dimensional spaces

What are the finite-dimensional spaces $W$ of differentiable functions with this property: If $f$ is in $W$, then $\frac{df}{dx}$ is in $W$.
2
votes
1answer
31 views

How to guess 2nd order ode solution

I am solving this equation $$ y''+y'\cos x-y\sin x=0 $$ My question is this: do i have to guess some y solution in order complete my answer with $y=y_1c_1+y_2c_2$, or is there some other way? If i ...
1
vote
1answer
22 views

First order nonlinear ODE

I'm looking for help to solve the following equation. The $y^2$ term is really confusing me. I suspect I need a substitution but cannot think what... $$x^2 y' = 1 - x^2 + y^2 -x^2 y^2$$ Many Thanks ...
0
votes
1answer
17 views

Inhomogeneous second order DE

When my second order DE is inhomogeneous, it has a $f(x)$ on the right hand side... I know that if $f(x)$ is of the form $f(x) = a^x * p(x)$ then my "guess" of a particular solution that solves ...
0
votes
2answers
29 views

Is there any other solution for this differential equation?

I have a differential equation $\sqrt{f(x)}=f'(x)\sqrt{x}$. Is there any other solution for this differential equation other than $f(x)=x$? Thanks very much.
0
votes
1answer
35 views

chain rule on trigonometric function

$y=(g(\sin(3x)))^4$ and $g(0)= 3$ , $g'(0)=\frac{1}{9}$. We are supposed to find the derivative at $x=0$. for $y'$ I tried and got $$y'=4(g(\sin(3x)))^3 \cdot ( g'(\sin(3x) + g(3\cos(3x))$$ What am ...
0
votes
1answer
28 views

What does this mean? Second order DE

Given the equation $y" + qy' + py = e^{ax}$ I need to find the complete solution for all a's. I've calculated the solutions for the homogenous differential equation, just not sure what my next move ...
0
votes
1answer
13 views

Applying conditions

I have this $\displaystyle 2(y')^2=2\cdot\sqrt y+c$ equation and $y(0)=1$, $y'(0)=1$. How do you tell from the information given, that $c$ here is equal to $0$? I just can not figure it out, thanks ...
0
votes
0answers
45 views

Condition for solutions to differential equation to be defined in $\Bbb R$

Let $f: \Bbb R \times \Bbb R^n \to \Bbb R^n$ a $C^1$ function. If there is a function $h: \Bbb R \to [0, + \infty)$ such that $\left\lVert f(t,x) \right\rVert \leq h(t) \left\lVert x \right\rVert $ ...
0
votes
1answer
30 views

$2^{\mathrm{nd}}$ order nonlinear ODE: $4y''\sqrt{y}=1, y(0)=1, y'(0)=1$

I am solving this second order nonlinear equation, that is in the title. My solution is: $$ \frac{4}{3}(y^{1/2}+c)^{3/2}-4c(y^{1/2}+c)^{1/2}+a=x $$ where $c$ and $a$ are constants that spawned ...
0
votes
0answers
11 views

reduction of order

I come across this equation in book $$F(z)=(1-\lambda + \mu )f(z) + (\lambda - \mu) zf'(z) + \lambda\mu z^2f''(z)$$ My question is how to reduce the 2nd order to 1st order? Can show some steps? ...
0
votes
1answer
44 views

Uniqueness of solutions to nonlinear ODEs

For a linear non-homogeneous ODE $$ \dot{x}=f(t),\ \ x(0)=x_0, 0\leq t\leq 1, $$ if one has two different solutions then one has infinitely many solutions. The idea for showing such property is to use ...
0
votes
1answer
52 views

Numerical way to deal with Dirac delta.

I have been wondering about this: I have a differential equation $y'(t) = y(t) + n \delta(t) y(t)$ with $y(-1) :=y_0$ Thus I want to apply a short delta pulse at some particular point $0$ to my ...
0
votes
1answer
33 views

Fuchs' theorem and ODE series solution

I have just learned to solve ODEs via power series method. I am a bit confused about the Fuchs' theorem. The theorem mentions ordinary and regular singular points at $x_0$ and our ability to find the ...
1
vote
2answers
22 views

Show that there are constants $K$ and $\alpha$ such that $|(e^{At})_{ij}|\leq e^{-\alpha t}K$.

I want to prove that if all eigenvalues of $\textbf{A}$ in the sytem $\dot{\textbf{x}}=\textbf{Ax}$ have negative real parts then there exist constants $K$ and $\alpha$ such that ...
3
votes
1answer
41 views

Gronwall type inequality

Is there a Gronwall-type inequality for bounding $u(t)$ such that $$\vert \partial_t u(t)\vert\leq C \{ u(t)+u(t)^\alpha\}$$ with $\alpha>1$ ?
1
vote
1answer
57 views

Inhomogeneous modified Bessel differential equation

I'm trying to solve the following inhomogeneous modified bessel equation. $$y^{\prime\prime}+\frac{1}{x}y^{}\prime-\frac{x^2+4}{x^2}y=x^4$$ I know the homogeneous solution for this differential ...
1
vote
1answer
42 views

Forced oscillation in a pendulum and displacement over time

At the old Exploratorium in San Francisco there used to be a Resonant Pendulum. A weak magnet tied with a string was used to exert a force on a steel plate wrapped around a 200kg concrete cylinder ...
0
votes
1answer
25 views

Classifying boundary conditions when PDE is given on the whole space.

I am being asked to classify the boundary conditions for: $u_{x} + u_{y} = 0$ such that $u(x,y)=1$ whenever $x=y$. I have only learned about three different boundary conditions: Dirichlet, Neumann, ...
1
vote
1answer
56 views

Autonomous differential equation

Let $f: \Bbb R \to \Bbb R$ and $x_0 \in \Bbb R$, such that $f(x_0)> 0 $, and assume that $x(t)$ is the solution of $x'=f(x)$, such that $x(0)=x_0$. If $f(x) > 0$ then $x(t)$ is defined for all ...
3
votes
1answer
29 views

Separation of variables - “formal” notation?

When using separation of variables technique to solve differential equations, I sometimes have both f(x) and g(y) on the right side, and then I divide by g(y) to separate them.... but how can I ...
1
vote
1answer
89 views

Solving differential equation.

In my research work I need to find the solution of the following differential equation. $\displaystyle y'(x)=\frac{y(x)+1}{2 \sqrt{x y(x)}-x},$ $y(0)=0$, where the solution must satisfies the ...
0
votes
1answer
30 views

Am I wrong with this simple differential equation?

I have been given the following differential equation: $$(x^2 + y)dx - xdy = 0.$$ The equation turns out to be inexact, so I opted for the simple straightforward solution with the integrating factor ...
0
votes
2answers
36 views

Find the general solution to differential equation

Could someone please help me with this if $\frac{dx}{dt} = 2x$ and $x(1)=x(0)+1$ find $x(t)$. I started off with: $$\frac12 \int \frac1x \, dx = \int 1\, dt$$ $$\frac12 \ln(x) = t+c$$ ...
1
vote
0answers
34 views

Solution to non-linear OIDE

How do I go about solving this equation? $\frac{\partial F(r,y)}{\partial r} = Q(r,y) - P(r,y) F(r,y) - R(r,y)F(r,y)\int_0^\infty dy'S(r,y') F(r,y')$ with the initial condition that $F(r=0,y) = 0 \ ...
0
votes
1answer
33 views

Existence and uniqueness of an ODE

I have the following ODE: $y'=y+e^{-2t}y^2$ I know the solution is $y=\frac{1}{e^{-2t}+ce^{-t}}$, c constant. Then the problem says that $y=0$ is a different solution. How can I explain that this ...
2
votes
1answer
24 views

Rewrite a differential equation formula

I am currently reading Elementary Differential Equations and I don't quite understand how they rewrote this Differential equation. I know that it is a very simple answer, but for some reason I can't ...
0
votes
0answers
19 views

The difference between euler line and direction field?

Can someone please explain to me what the difference is between the direction field and euler lines? Direction field is the tangent line to the integral curve of a differential equation, as far as I ...
0
votes
0answers
18 views

Finding solutions for a nonlinear ODE of the second order

Try to solve in closed form the ODE (any kind of special functions can be used, or a solution in parametric form is also valid) yy'' + y' = x, (1) y(x_0) and y'(x_0) are open for any x_0. Note ...
2
votes
1answer
143 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation $$ f^{(n)}=f, $$ and with $\,f,f',\dots,f^{(n-1)}\,$ being linearly independent. ...