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

learn more… | top users | synonyms (1)

0
votes
2answers
19 views

Method of Undetermined Coefficients - Efficient Way of Finding Constants

$$y'' - 4y = (x^2 - 3)\sin(2x)$$ I have to solve the following differential equation. I have gotten everything setup. I have two questions however. Is the form of the particular solution: $y_p = ...
0
votes
0answers
13 views

Prove existence of unique maximal solution

I need to prove that $x'=t^2\sqrt{1+2x},~ x(0)=0$ has a unique maximal solution. I thought of using that $\sqrt{1+2x}$ is bounded by a linear function, but the only theorem I know about that needs ...
0
votes
0answers
22 views

Classification of Differential Equations

I know that there is a theory of integrating (partial) differential equation by finding its symmetries (which form a Lie group) and making corresponding transformation of the domain. I also know ...
1
vote
2answers
32 views

Periodicity of trigonometric functions directly from their power series

My question is very simple yet I've gotten nowhere with it. Is there any way one can, without directly or indirectly referencing any differential equations satisfied by the circular trigonometric ...
0
votes
3answers
22 views

Another confusion about initial condition in ODE

I am still having some confusion on certain problems. Here is what I mean, the questions asks to solve the IVP and determine how the interval on which the solution exists depend ends on the initial ...
1
vote
2answers
18 views

Brief question in regard to existence and unique (ODE)

I am just having a bit of trouble understanding the answer to this problem. It asks where in the ty plane would the ODE satisfy the existence and uniqueness theorem, that is; $\mathbf{Thereom}:$ ...
0
votes
3answers
33 views

Algebraic Manipulation of a Separable Equation

$$(sin(x)+x^2e^y-1)dy/dx=-ycos(x)-2xe^y$$ I understand how to do these problems but I'm wondering how do I get the y variables on the left hand side and the x variables on the right hand side so that ...
2
votes
0answers
29 views

Prove the $f_1, f_2$ is a basis of linear subspace of solution of differential equation

Let $p,q \in C(\mathbb{R}), L_{pq} = \{f \in C^2(\mathbb{R}):f^{(2)} + pf^{(1)} +q f = 0\} $ For each $(a,b)^T \in \mathbb{R}^2$ there is only one $f \in L_{pq}$ with $(f(0),f'(0)) = (a,b)$ 1- ...
0
votes
1answer
28 views

How could we continue to show the inequality?

Let $\Omega$ a bounded space. Let $u_1$ the solution of the problem $$-\Delta u_1(x)=f(x), x \in \Omega \\ u_1(x)=g_1(x), x \in \partial{\Omega}$$ and $u_2$ is the solution of the problem $$-\Delta ...
1
vote
3answers
37 views

solving second order non-homogeneous differential equation 2

I have a problem on solving this differential equation. $ y=c_{1}+c_{2}e^{-2x}-cos2x-1/2(sin2x-cos2x) $ I reached : $ y''+2y'=4sin2x $ as answer but I'm not sure. my step by step solution is : $ ...
0
votes
1answer
34 views

Determining the maximum intervall of existence

Can somebody explain to me the concept of an intervall of existance/interval of validity? Is it basically the domain of my differential equation? I tried to look it up and I came across this site ...
2
votes
1answer
22 views

$y''(x)+2 x y'(x)+\left(x^2+1\right) y(x)=0$

I constructed this equation so that it would have a double root for $e^{-x^2/2}$. I basically applied $(D+x)(D+x)y$, which gave me this equation. The solution is $c_1 e^{-\frac{x^2}{2}}+c_2 ...
1
vote
1answer
37 views

Converting a non-linear ODE to a Bernoulli equation

I am self-studying differential equations using MIT's publicly available materials. The first part of one of the recitation exercises runs as follows: Show that \begin{align} (3e^{2y}x^{\frac{2}{3}} ...
2
votes
1answer
19 views

Find minimizer of the functional

Find minimizer of the functional $ l(u)= \int \limits _{-1} ^1 u(t) \mathbb d t $ with $u(-1)=u(1)=0 $ subject to $g(u)=\int \limits _{-1} ^1 \sqrt{1+u'(t)} \mathbb d t=π $. I solved it using ...
1
vote
1answer
34 views

How could we continue to get a contradiction?

Let $\Omega$ a bounded space. Using the maximum principle I have to show that the following problem has an unique solution. $$-\Delta u(x)=f(x), x \in \Omega \\ u(x)=g(x), x \in \partial{\Omega}$$ ...
-2
votes
1answer
54 views

Find the general solution of the differential equation $\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$ [closed]

Find the general solution of the differential equation $$\left(3y^2+x^2+x+2y+1\right)\cdot y'+2xy+y=0$$
1
vote
3answers
53 views

I've got the following differential equation, how do I integrate the expression to get the answer?

-Solve the differential equation ,with the given condition: $${\partial z \over \partial x}+(2e^x-y){\partial z \over \partial y}=0.\ \ z=y\ \ \ at \ \ \ \ \ x=0. $$ I solve it as follows: $${dx ...
-1
votes
2answers
37 views

Find the general solution of the differential equation $y' (y+x)=y $ [closed]

Find the general solution of the differential equation $$y'(y+x)=y. $$
-5
votes
1answer
35 views

Find the integral curve of the differential equation $y'-\frac{2y+1}{x}=1$ belongs point of $M\left(0;-\frac{1}{2}\right)$ [closed]

Find the integral curve of the differential equation $$y'-\frac{2y+1}{x}=1$$ belongs point of $$M\left(0;-\frac{1}{2}\right)$$
2
votes
3answers
50 views

How would this differential equation be solved?

How would this differential equation be solved? $$y{\partial z\over \partial x}+z{\partial z\over \partial y}={y \over x}$$ I was taught to solve them like : $${dx \over y}={dy \over z}={dz ...
1
vote
2answers
32 views

Solving linear differential equation $t\dot{x}(t)+3x(t)=-\frac{1}{t^2+1}$

I want to solve the following linear differential equation (initial value problem): $$t\dot{x}(t)+3x(t)=-\frac{1}{t^2+1}; x(1)=\frac{\pi}{4}$$ I first tried to solve the homogeneous diff. eq. ...
0
votes
1answer
31 views

Problem on string vibration

Given the standard wave equation for small amplitudes, we have been asked to find the position of string $y(x,t)$, given: $y(x,0)=\sin x$, and, $y'(x,0)=\cos x$, where $y'$ depicts partial ...
0
votes
2answers
37 views

solving second order non homogeneous differential equation

I am given a non homogeneous differential equation $$ y''+4y = 3 \csc 2x. $$ When I try to find value of the Wronskian $W(y_{1},y_{2}),$ the result is zero. I can't solve it the differential ...
2
votes
2answers
56 views

Minimizing a functional with a free boundary condition

Find the extremals of the functional $$\text{J}(y)= y^2(1) + \int_0^1 y'^2(x)dx , \ \ y(0)=1.$$ Answer: $y(x)=1-\frac{1}{2}x$ My solution: $ F (x,y,y')=y'^2(x)$ After solving the ...
3
votes
1answer
51 views

Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
4
votes
1answer
77 views

Backwards Heat Equation $ u_{t} = -\lambda^2 u_{xx}$

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} & = \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
4
votes
1answer
116 views

Nonlinear Partial DE

In my work I have faced with following partial differential equation $$\left(\frac{\partial u}{\partial x}\right)^2-\left(\frac{\partial u}{\partial y}\right)^2+f(x,y)\frac{\partial u}{\partial ...
1
vote
1answer
30 views

Is there a way to solve systems of linear differential equations without using eigenvector/eigenvalues?

I'm teaching basic differential equations this semester and I was wondering if there were methods of solving systems of linear differential equations that don't use (directly) ...
2
votes
0answers
27 views

$V$ is $C^1$ and $V(x_0)=0$ and $ \nabla V $ is not zero $\{ x : V(x)= c \}$ is a surface with no edge around $x_0$

I am studying lyapanov second method in stablity theory of ODE. I have encountered a geometric lemma which says the following: Assume $ V:\mathbb R^n \to \mathbb R$ is a $C^1$ and $x_0 \in \mathbb ...
1
vote
1answer
31 views

Solving differential equation $\dot{x}(t)=c\cdot x(t) \cdot (1-x(t))$

I am pretty new to differential equations and I am having trouble solving the following equation: $$\dot{x}(t)=c\cdot x(t) \cdot (1-x(t)); \space x(0)=x_0$$ I tried separating the variables: ...
-1
votes
1answer
43 views

How to find the general solution of the following differential equation [closed]

Could someone please explain to me how to solve the differential equation below: \begin{equation*} 2y\cot x\frac{dy}{dx} = (4+y^2)\cos x? \end{equation*} Thank you very much :)
0
votes
1answer
25 views

A simple version of the Picard-Lindelöf Theorem

I wish to ask a particular question about following the proof in this theorem, and thought the best place to come might be here. It is as follows: First, we have a differential equation that ...
1
vote
0answers
33 views

Theorem to show trajectories of differential equations are close after small change to initial condition

Consider two solutions(or trajectories), say $x_1(t)$ and $x_2(t)$, of a system of differential equaions. That is, $$ x_1'(t)=x_2'(t)=f(x,t), t\ge0. $$ Also, $\|x_2(0)-x_1(0)\|<\epsilon$ for some ...
0
votes
1answer
36 views

Help in solving this differential equation.

I'm trying to solve this differential equation but having some trouble as there is a constant in there which would change the solution depending on its value: ...
3
votes
2answers
63 views

Solve first order nonlinear differential equations

I want to solve this nonlinear 1-st order ODE, $$\frac{1}{1+x}=(\frac{1}{x-y}-\frac{1}{y})\frac{dy}{dx}$$ I find it non-separable, and Wolfram Alpha does not give me a closed form solution, but the ...
1
vote
2answers
60 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
0
votes
0answers
17 views

An application on Sturm-Liouville Theorem

Consider $$((1 + x^2)y')' + λy = 0$$ with $y ' (0) = 0 , y(1) = 0$. How Sturm-Liouville Theorem implies that this boundary value problem has no solution when $λ < 0$ and $|λ|$ is large.
0
votes
1answer
42 views

Confused about solving general second order linear ODE

I am having some difficulty having a more deep understanding of solving general ODE, I will give an example because it is hard to explain without me showing some work. Find the general solution to ...
0
votes
1answer
45 views

Trouble with Wronskian

I am having some trouble justifying to myself something. Please see my other question in which I talked about solving if the functions f and g are linearly independent or not where $$f(x)=\cos(3x)$$ ...
0
votes
1answer
46 views

An ODE problem that seems weird!

Well, as you can see in the image above, the problem consists of a Linear-Non Homogeneous ODE. It goes on and tasks the student with transforming such ODE into a Linear 4th Order ODE containing only ...
3
votes
4answers
259 views

Proving a function is not differentiable

Given the function $f(x) = |8x^3 − 1|$ in the set $A = [0, 1].$ Prove that the function is not differentiable at $x = \frac12.$ The answer in my book is as follows: $$\lim_{x \to \frac12-} ...
-1
votes
3answers
40 views

Laplace transform of f(x)

Let $\mathcal{L}$ be the Laplace transform and $$\mathcal{L}\{f(t)\} \cdot (s^3+s-1) = \frac 1 {s-1}$$ I am trying to find $f(t)$, it's complicated!
0
votes
2answers
26 views

Is the Wronskian determinant positive or negative?

The Wronskian of the general solution $y(x)=ay_1(x)+by_2(x)+cy_3(x)$, to a third order differential equation, is given by $$ W = \begin{vmatrix} y_1 & y_2 & y_3 \\ y_1' & y_2' & y_3' ...
0
votes
1answer
31 views

System of differential equations using Laplace transform

Using Laplace transform, solve the system: $w'+y=\sin(x)$ $y'-z=e^x$ $z'+w+y=1$ where $w(0)=0$ and $z(0)=y(0)=1$.
2
votes
1answer
25 views

How can I find an equation of motion of this?

There is a point of mass moving in the xy-plane with harmonic forces acting in x- and y-direction $F_x=-m\omega^2x$ and $F_y=-m\omega^2y$. At the same time there is an additional force acting in the ...
0
votes
1answer
25 views

Can't understand a notation regarding weak solution of Vlasov-Poisson system

The text is from https://cmouhot.files.wordpress.com/2010/01/chapter5.pdf . In section 1, it uses $~f_t~$ to represent a smooth solution of Vlasov-Poisson system (VPS). I think here $~t~$ in $~f_t~$ ...
1
vote
1answer
54 views

Prove a function has a maximum and minimum along a domain

Given the function $f:[13,132] \to R$ defined by $f(x)=sinx+x^3-$2 $e^x $ prove that the function has a maximum and minimum along the domain. I understand that a function has a maximum and minimum ...
0
votes
0answers
29 views

A Boundary Value Problem $y^{''} + xy = 0, x \in [a, b]$

The following problem is an exam problem, which I could not do. But I attempted it as far as I can do. Let $y$ be a nontrivial solution of the boundary value problem $$y^{''} + xy = 0, x \in [a, b], ...
0
votes
3answers
222 views

Series Solution of an ODE

The ODE below is required to help compute the coefficients of function. There isnt any information about this topic in my textbook so i am just wondering how i would go about this question? In this ...
0
votes
3answers
40 views

Analytically solving nonlinear second order ODE

I need help with providing an answer to this nonlinear ODE $a_1 + f_1(x) + f_2(x) y' - a_2\bigg((y')^2 - y''\bigg) = 0,$ where the $a_i$'s are constants and the $f_i$'s are arbitrary functions of ...