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

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-2
votes
0answers
31 views

How to find the required differential equation [on hold]

How to find the differential equation of tangent lines to the parabola y=x^2? How to find the differential equation of all conics whose axes coincide with axes of co ordinates? I think the equation ...
1
vote
2answers
51 views

homogeneous differential equations $y' = f(y/x)$

There is a weird Theorem that comes about when considering whether a function is homogeneous (in the sense of the title definition). I was unable to prove it, or to find a proof to it. Can any one ...
1
vote
1answer
42 views

Solve $A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$

a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE $$A \partial_t w + B \partial_t\partial_x^4 w ...
2
votes
1answer
799 views

Wrong answer for this differential equation temperature problem.

(a) An object is placed in a 68°F room. Write a differential equation for H, the temperature of the object at time t. ANSWER: dH/dt = -k(68 - H) (b) Give the general solution for the differential ...
0
votes
1answer
19 views

How much can I gauge about the domain of a differential equation without actually solving it?

Say I have the differential equation $$y' = \frac{3t^2 - 2ty}{4 - t^2} \text{, where }y(1)=-3$$ Clearly the equation is undefined at $t = \pm2$, and a solution exists at $t = 1$. Can I conclude from ...
3
votes
1answer
30 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
1
vote
2answers
63 views

Unable to solve $y''+\lambda y =0$

I wish to find the eigenvalues and eigenfunctions of the following, but am unable to and further don't know what I am doing wrong at all $y''+\lambda y =0$ where $y(0)=0$, $y'(1)+y(1)=0$ My ...
3
votes
3answers
45 views

A simple problem on first order differential equations

An ODE (Ordinary Differential Equation) of order $n$ becomes a relation: $$F(x,y,y^{(1)},...,y^{(n)})=0$$ Then $F(x,y,y^{(1)})=0$ defines an ODE of order one. In "basic standard texts", for purposes ...
0
votes
1answer
43 views

Path to Self Study Calculus 1-4 and Linear Algebra [on hold]

For the past year I've taken up self studying mathematics. My initial intent was to study so that when I entered college (currently a junior) I would have most of the basic mathematics for studying ...
0
votes
4answers
56 views

Show the given series is a solution of $y''-xy'-y=0$

My problem is this: "Show that the function represented by the power series, $$y=\sum_{n=0}^{\infty} \frac{x^{2n}}{2^nn!}=1+ \frac{x^2}{2}+ \frac{x^4}{8}+ \frac{x^6}{48}+...$$ is a solution of the ...
1
vote
1answer
22 views

What does it mean for a coefficient to have singularities?

I'm reading a paper that mentions ordinary differential equations with singular coefficients. I'm not sure what the author means by "singular coefficients". The paper is about uniqueness results for ...
1
vote
1answer
45 views

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$ ? Is this just a different notation?

Is $(x^2+y^2+x)dx+xydy$ the same as ${dy\over dx}(x^2+y^2+x)+{dx\over dy}(xy)$? Is this just a different notation?
9
votes
1answer
110 views

Solve the given Differential Equation

Solve the equation $$\frac{dy}{dx}=\frac{x+2y-5}{2x+xy-4} $$ I tried substituting $x=X+h$ and $y=Y+k$ but the $xy$ term is creating problem. How to solve it?
3
votes
3answers
66 views

solution of differential equation $\left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$

The solution of differential equation $\displaystyle \left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$ $\bf{My\; Try::}$ Let $\displaystyle \frac{dy}{dx} = t\;,$ Then Diferential equation convert ...
31
votes
2answers
845 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
-1
votes
0answers
33 views

Differential equation $2xy-\sin(x)+(x^2+e^x)y'=0$ [on hold]

can i get help with this ED? $$2xy-\sin(x)+(x^2+e^x)y'=0$$ Thanks in advance
3
votes
0answers
47 views

Resolving ODE-1 $(x^2 + y^2 +x)\,dx + xy\,dy=0$ am I wrong or my teacher is?

This is how I've resolve this ODE-1 : $$(x^2 +y^2 +x) \, dx + xy \, dy=0$$ Check if the eq is exact: $${\partial M \over \partial y}={\partial \over \partial y}(x^2 +y^2 x)=2y$$ $${\partial N \over ...
1
vote
2answers
30 views

Another Differential Equation

Having trouble (again) with this DE can someone help me find the general solution for it? I feel like my biggest problem is doing the algebraic manipulations to identify what kind of DE it is. $$y' + ...
1
vote
2answers
31 views

Differential Equation $y' = 2y/x - 1$

Can I get help solving this DE? $$ y' = \frac 2xy - 1$$ Doesn't look too hard but i just can't get to the correct result. Thank you in advance
4
votes
3answers
165 views

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$

For this problem, $y$ is a function of two variables: one space variable $x$ and one time variable $t$. $k > 0$ is some constant. And $x$ takes is value in the interval $[0, 1]$ and $t \ge 0$. At ...
0
votes
0answers
17 views

Show that the function is identically Zero in certain subset

We are given a open ball D (radius = 1) in $\mathbb R^2$. and let $\{x_n\}$ be the dense sequence in the set D. Around each point $x_n$ we make a hole of radius $r_n$. The sequence $r_n$ satisfy the ...
1
vote
0answers
41 views

Acceleration of an air bubble under the sea

An air bubble arises from the bottom of the sea. Find its acceleration if the resistance force is proportional to $\rho$*A*$v$ where $\rho$ is density of water, A is cross section area and $v$ is ...
1
vote
1answer
62 views

When does the same trajectory appear in two dynamic systems from the same point?

Imagine you have two dynamical systems, given by the statespace equations: $\frac{dx}{dt}=F_1(x)$ and $\frac{dx}{dt}=F_2(x)$, and you are concerned with trajectories form a point in phase space $x_0$. ...
1
vote
0answers
29 views

In a recurrence relation, how do we know which order to terminate?

By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish. For example, in ...
1
vote
1answer
51 views

Reachability from non-zero initial state?

I have the following system: $$ \dot x(t)=\begin{bmatrix} -2 & 1 & 2 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}x(t)+\begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}u(t) $$ The ...
0
votes
1answer
16 views

How to solve a boundary value problem of a Laplace equation?

Suppose $x,y$ are in the range $0 \leqslant x \leqslant 2,0 \leqslant y \leqslant 1$, I can use separation of variables to get $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ...
0
votes
2answers
24 views

Derivation of the variation of parameters in Second-Order Differential Eq.

In Second-Order ODEs ,There is a problem which I haven't solved. Method of Variation of Parameters; In derivation of the method , there is a part which is following ; P and Q just constant. ...
3
votes
1answer
182 views
+50

Biharmonic Equation in a Rectangle with Some Uncommon Boundary Conditions

Consider the following boundary value problem (BVP) $$\matrix{ {{\Delta ^2}H = 0,} \hfill & {} \hfill & {{\rm{in}}\,} \hfill & \Omega \hfill \cr {\partial _y^2H = 0} \hfill & ...
0
votes
1answer
27 views

Approaches to stability of newtonian systems

I am having some difficulties figuring out how to approach "Test stability problems". I usually test the linearization of the system (since it is very straightforward and easy), and if that doesn't ...
0
votes
0answers
22 views

Does a Liapunov function h to have all the variables explicitly?

If I have for example, a system like this $$\begin{matrix} \dot{x}=f(x,y) & \\ \dot{y}=g(x,y) & \end{matrix}$$ in which i have to prove stability using a Lyapunov function. Now, if i have ...
0
votes
1answer
31 views

First Order Differential Equation for a Harmonic Oscillator

A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$. Now I have the following ...
1
vote
0answers
37 views

perturbation of exponentiolly stable system

consider the following system on $\Bbb{R}^n$ $\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $ assume that f(0,t)=g(0,t) = 0 and 1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$ ...
2
votes
3answers
222 views

Solution of differential lyapunov equation

How would I solve for following, else any implemented algorithms or solvers in matlab (even ways to solve it) for Lyapunov differential equation of form: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
1
vote
1answer
18 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
2
votes
1answer
28 views

Differential equation$ (x^2-x)y' = (y^2+y)$

Can i get help solving the differential equation $$y' = \frac{y^2+y }{x^2 -x}$$ I tried searching but could not find anything similar. Thank you!
0
votes
1answer
12 views

Maximally extended solution of this ODE.

So I am asked to find the positive, maximally extended solutions to this ODE. $$u'(x) = \frac{x}{u(x)}$$ Now a solution is given by $$u(x) = (\int_{y_0}^y t dt )^{-1}\circ \int_{x_0}^x s ds = ...
1
vote
3answers
35 views

Differential equation $2f'(t)+tf(t)=0$ with $f(0)=\sqrt{\pi}$.

How to solve the following differential equation: $$2f'(t)+tf(t)=0$$ with $f(0)=\sqrt{\pi}$. I tried to write $2f'(t)+tf(t)=0$ something like $(f(t)g(t))'=0$ for some function $g$ but it was ...
0
votes
1answer
11 views

Expressing $x$ and $z$ as functions of $y$ (non-generate matrix)

Consider the system $$ \dot{x}=x-z+y^2,\quad\dot{y}=x-2y+z+y^2+2x^2z,\quad\dot{z}=-2x+2y+z^2-y^2. $$ and the equilibrium $(0,0,0)$. Now, there is used some statement that I did not know yet: ...
2
votes
1answer
35 views

Prove that if $A\neq B$ then $\exp(A/n) \neq \exp(B/n)$ for some $n\in \mathbb N$

Let $A \neq B \in M_{n\times n}$ be linear maps. I'd like to prove that there exists $n\in \mathbb N$ such that $e^{A/n} \neq e^{B/n}$. I tried assuming that $e^{\frac{A}{n}} = e^{\frac{B}{n}}$ for ...
1
vote
1answer
25 views

Why is this system reversible? What does this mean?

Consider the system $$ \dot{x}=y,\qquad\dot{y}=-x+y^2. $$ Then, it is said that the system is reversible $(t\to -t, y\to -y)$. What does this mean? If I put this into the equations, I get $$ ...
10
votes
0answers
204 views
+50

If any differential equation is given by $f''(x)+f'(x)+f^2(x) = x^2\;,$ Then $f(x)=$

If any differential equation is given by $f''(x)+f'(x)+f^2(x) = x^2\;,$ Then $f(x)=$ $\bf{My\; Try::}$ We can write above differential equation as $$e^xf''(x)+e^xf'(x)+e^x\cdot (f(x))^2 = ...
1
vote
1answer
38 views

How to prove $\frac{dh}{dt}= \frac{5 }{h^2} - \frac{1}{20}$ and a couple other related questions (complete information inside)?

This is a differential equation question. Since I might not be able to explain is well, I will attach a link to the question as well as a screenshot of the mark scheme. Question: Mark Scheme: ...
0
votes
0answers
12 views

capillary surface problem [closed]

Consider the capillary surface problem (⋆) )   Du  div 1 + |Du|2 = κu in Ω on∂Ω,  Dηu  1+|Du|2 =β where κ > 0, η is the outward pointing unit normal to ∂Ω and β ∈ C1(Ω) satisfies |β| ≤ 1 ...
0
votes
1answer
26 views

Wolfram Alpha Step By Step For Systems of differential equation

Does anyone know if wolfram alpha has step by step solutions for systems of differential equations? When I input them, it comes up with an answer but it does not give me the step by step solution. I ...
0
votes
1answer
31 views

Help with this differential equation, nonlinear

How would I solve the following Differential Equation $\frac{dy}{dx}= \sqrt{x+y} $ Clearly, it is nonlinear and non homogeneous, I could not find the way to solve it with Bernoulli or to make it an ...
1
vote
1answer
38 views

Raising e to the power of both sides of an equation

I have a simple question: in differential equations, it has been common in several of my homework problems to raise a base $e$ to the power of both sides of an equation to get variables out of natural ...
0
votes
1answer
17 views

Differential equation with shifted argument.

What are the methods for solving the following class of problems: $$ \frac{df(x)}{dx}=a f(x-\xi), $$ or $$\begin{cases} \frac{\partial F(x_1,x_2)}{\partial x_1}=a_1F(x_1-\xi_{11},x_2-\xi_{12})\\ ...
-1
votes
0answers
48 views

Solving a system of nonlinear second-order differential equations with initial/boundary conditions.

I have developed a set of $n$ equations, $n$ variables for my dynamic system. The derivatives are second and first order in terms of $\theta$ (angle) of different components of the system (basically a ...
0
votes
1answer
31 views

Trapezoidal rule - truncation error

I am trying to prove that when solving numerically diff. eq.: $$ y'(t)=f(t,y(t)), \hspace{0.5cm} y(t_{0})=y_{0} $$ using trapezoidal rule, namely: $$ y_{n+1}=y_{n} + \frac{h}{2} \left( f(t_{n},y_{n}) ...
1
vote
0answers
23 views

Repeated Eigenvector/Eigenvalue matrix method

So I am having trouble with finding the generalized solution and I am not sure why my answer is interpreted as incorrect and I wanted to double check. $$ \overrightarrow{y'} = \begin{pmatrix} -6 ...