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

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0answers
30 views

Compute the operator norm of the linear transformation defined by the following matrix. [closed]

Compute the operator norm of the linear transformation defined by the following matrix. \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}
1
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4answers
64 views

Solving this 1st Order PDE [closed]

I am trying to solve the following PDE with an initial condition: $$u_x + u_y = x + y$$ with $$u(x, 0) = 0$$ I am not sure which method to use to solve this. Thanks
1
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1answer
625 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 ...
2
votes
1answer
23 views

How to prove that a differential equation has a solution

I want to prove that there exists $f : [0,1] \to [0,1]$ such that $$ w(y - f(y)) = \int_0^y g(x) dv(f(x)), $$ where $w : [0,1] \to [0,1]$ and $v : [0,1] \to [0,1]$ are continuous strictly ...
1
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1answer
69 views

Use of exclamation point

I'm quite puzzled by the use of an exclamation point in this paper. The authors introduce the following linear constraints to a quadratic program: $ \sum_k a^{(l)}_k b_j (\mathbf{x}_k) = r_j^{(l)} $ ...
1
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1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
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0answers
31 views

A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
1
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1answer
14 views

Various forms of the Confluent Heun Equation

The Confluent Heun equation is expressed in various forms. It's non-symmetrical canonical form is: \begin{equation} ...
1
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1answer
48 views

Solution space of the differential equation $y'' + y =0$

To find the dimension of the solution space of the equation given $y'' + y = 0$ . Take $y=e^{mx}$Then we have to solve the equation $m^{2}+1=0$ for $m$ . Which gives $m=\pm i$. ...
1
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1answer
46 views

System of differential equation with variable coefficent

How to solve this system of differential equations $x'(t)=\frac{a+s}{(1-t)d}x(t)-\frac{b}{(1-t)d}y(t)$ and $y'(t)=\frac{a}{(1-t)d}x(t)-\frac{(s+b+(1-t)c)}{(1-t)d}y(t)$ where a,b,c,d and s are ...
0
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1answer
19 views

Asymptotic error expansion of global error for single step methods

My question refers to the proof of the following theorem, but it may suffice to just skip the theorem and continue with the problematic taylor expansion $(\ast)$: Let $f(t,y)$ and the single step ...
2
votes
1answer
32 views

Solving this 2nd Order non-homogeneous PDE

I am trying to solve the following equation: $$3u_{xx} - 10u_{xt} - 3u_{tt} = \sin(x + t)$$ I know that the left hand side is a quadratic equation which I have to factorise. Then I let one of the ...
0
votes
2answers
65 views

find general solution to the Differential equation

Find the general solution to the differential equation \begin{equation} \frac{dy}{dx}= 3x^2 y^2 - y^2 \end{equation} I get \begin{equation} y=6xy^2 + 6x^2 y\frac{dy}{dx} - 2y\frac{dy}{dx} ...
1
vote
1answer
20 views

Where can I find the theorem that says an n order diffeq has n solutions?

I study engineering not mathematics and I feel my theoretical understanding of differential equations is SO wishy washy. I'm not interested in proving such a theorem at the moment, but I want to know ...
1
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1answer
32 views

Finding general solution to Partial Differential Equations

I am asked to find the general solution $f(x, y)$ of the partial differential equation: $\frac{\partial ^2 f}{\partial x \partial y}=e ^ {x+2y}$ I know these are relatively easy to solve, I haven't ...
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4answers
59 views

What is the essential difference between ordinary differential equations and partial differential equations?

Please forgive my stupidity. So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between ...
0
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0answers
42 views

Algebra behind Feynman-Kac theorem?

According to many sources, The Feynman-Kac theorem in Equation (1) below is the solution to Equation (3) - if X(t) follows a diffusion such as in (2). (Most Important) - Can someone show the algebra ...
0
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1answer
12 views

First order differential equation, getting from one step to another

I don't understand how to get the last equation, the first three are no problem, I just can't get the last equation from the third one.
1
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1answer
80 views

I just need your approval

Verify by susbtitution if the given functions are a solution to the next differential equations. a) $$x^2y''+xy'-y=\ln x \quad ,\quad y_p=x^{-1}-\ln x $$ simplifying: $$ x^2y''+xy'-y=\ln x $$ $$ ...
0
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3answers
33 views

Non-Linearity of a First Order ODE

I was working on the classification of ODEs as linear and non linear. For the equation $\ \frac{dy}{dx} = \frac{y(2-3x)}{x(1-3y)}$ I understand that it is a first order ODE where y is the independent ...
0
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1answer
30 views

Can there be different values of $y_p$ for one equation?

For example, consider following example: Solution given by book is this: I solved it using different approach(as shown in the pic below) & got different answer. Is my solution wrong or ...
0
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4answers
91 views

How did author do this algebraic manipulation?

In some question of differential operator author did this algebraic manipulation without explanation. Can someone explain this? Did he do it by completing the square or $$(a^3 + b^3) = (a+b)(a^2 - ...
2
votes
3answers
50 views

Given $\dot x(t)=3x(t)$ and $x(0)=\frac32$ what is $x(2)$?

I believe this is a differential delay equation but I'm not sure. I've tried integration but that was confusing. It has also been suggested that I can find the original equation from the first ...
3
votes
1answer
38 views

need help to understand the differential equation

In one of the books, it was mentioned $\frac{d}{dx}(x^3 \tan x)= (x^2\sec^3x+3x^2\tan x)$, but i think it should be $(x^3\sec^2x+3x^2\tan x)$. I feel its a printing mistake. Just wanted to be sure, ...
4
votes
0answers
30 views

Finding intersections of tori/toruses

I am looking for intersections of three tori. Is this possible? If so, how? To put things in perspective: I am looking for the coordinates of point P in space, and I have a triangle on the 'ground'. ...
0
votes
1answer
45 views

Help solving this nonlinear first order differential equation?

$$(y')^2=y^2 f(x)+1$$ I know that $y'',y'\gt 0$ and $y$ is defined on $[0,\infty)$. I tried doing an asymptotic analysis where the 1 is trivial so we can take the square root of both sides and then ...
4
votes
1answer
525 views

Software for numerical solution of a non-linear ODE system?

I have been given a nonlinear system of ODEs which has arisen out of a colleague's engineering research: $$\begin{array}{rcl} \dot{x}_0&=&x_1\\ ...
2
votes
2answers
55 views

Which of these 1-D representations of the Navier-Stokes equations is correct?

The incompressible Navier Stokes equations can be written as A. $$\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = S$$ or B. $$\frac{\partial ...
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0answers
18 views

Parabolicity of high order PDEs

I know that the traditional classification of PDEs into parabolic, elliptic, and hyperbolic is applicable for the second order equations. However, I often see remarks about parabolicity of higher ...
2
votes
2answers
50 views

Find the Integrating Factor

Show that the differential equation $(1-2x^2y^2-4xy^3)dx + (2-2x^3y-4x^2y^2)dy=0$ is not exact, but admits integrating factor $\mu=\mu(xy)$. Find $\mu$ and solve the equation. With the method I ...
0
votes
1answer
41 views

How can I find the critical curves for the following functional

Find the critical curves for the following functional : $$J[y,z]=\int_{0}^{1} \sqrt{1+y'^2+z'^2}$$ such that :$$y^2+z^2=1$$ and $$y(0)=z(1)=1$$ $$y(1)=z(0)=0$$
2
votes
2answers
53 views

Verify that $y=x^3+7$ is a solution of the differential equation $y'=3x^2$

I know this is trivial but I don't don't know if I'm right Verify, by substitution, if the function is a solution of differential equation. $y'=3x^2$ , $y=x^3+7$ Differentiating the function ...
0
votes
1answer
27 views

Convergence of Linear First-Order Differential Equations

Suppose $u$ is a twice continuously differentiable function with linear growth, $$\lim_{x\rightarrow \infty} u'(x)-\frac{1}{g(x)} u(x) = 0 $$ and $g$ is a Lipschitz continuous function with Lipschitz ...
4
votes
2answers
64 views

Existence of a solution for a nonlinear ODE on $[0,\infty)$

I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by $$ \begin{cases} y'(t) = y^2 \cos(t)-ye^t \\ y(0)= y_0 \end{cases} $$ where $y_0 ...
5
votes
1answer
50 views

Solve the PDE: $u_{xx} - 3u_{xt} - 4u_{tt} = 0$

It is asked to solve the PDE $$u_{xx} - 3u_{xt} - 4u_{tt} = 0$$ using a factorization, that consists in $$\left( \frac{\partial}{\partial x} - 4 \frac{\partial}{\partial t} \right) \left( ...
-2
votes
1answer
54 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 :)
2
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1answer
39 views

Function of single variable $f(x)$, $f(x+y)=f(xy)$ and the exponential.

For a function of a single variable $f(x)$ which has the property that $f(x+y)=f(x)f(y)$ we first set $x=y=0$ and then develop an ODE to show that $f(x)=\exp\{-\beta x\}$. I do not understand how ...
2
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1answer
41 views

Simultaneous Differential Equation $\frac{dx}{x}=\frac{dy}{x+y+t}=\frac{dt}{t}$

$$ \frac{dx}{x}=\frac{dy}{x+y+t}=\frac{dt}{t} $$ I was given this problem but don't know how to start, I'm not sure but I tried it like $$\frac{dx}{x}=\frac{dy}{x+y+t}\tag{1}$$ ...
2
votes
1answer
33 views

To find Inverse Laplace of $\,F(s)=\log\dfrac{s+1}{(s+2)(s+3)}$

To find Inverse Laplace of $$F(s)= \log\frac{s+1}{(s+2)(s+3)}.$$ I have tried to use shifting theorems, but of no use. Should I apply series for log and take inverse laplace of individual terms, if ...
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0answers
42 views

A question about an unusual ODE [closed]

Is it possible to solve this $$y'=\frac{y}{2x} +x\sin\left(\frac{y}{x}\right)\ \text{?}$$
0
votes
2answers
22 views

Simultaneous D.E. $2\frac{dx}{dt}-x+\frac{dy}{dt}-2y=0$ ; $3\frac{dx}{dt}-2x+2\frac{dy} {dt}-3y=0$

$$ 2\frac{dx}{dt}-x+\frac{dy}{dt}-2y=0 \tag{1} $$ $$ 3\frac{dx}{dt}-2x+2\frac{dy} {dt}-3y=0 \tag{2} $$ If you subtract $\,(1)\cdot2 - (2)\,$ I get $\,\dfrac{dx}{dt}-y=0\,$ which does not make sense ...
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0answers
13 views

Which statement would be correct: “All homogeneous ODEs are also an exact ODE” or “All exact ODEs are also a homogeneous ODE”?

I was asked by my math teacher, but so far couldn´t find the correct answer "All homogeneous ODEs are also an exact ODE" (A) or "All exact ODEs are also a homogeneous ODE" (B) ? Someone knows any ...
0
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1answer
21 views

How is implicit ODE different from explicit ODE?

I've seen the wikipedia article; there it has been written as : Let F be a given function of x, y, and derivatives of y. Then an equation of the form $$F\left(x,y,y',y'',\cdots,y^{(n-1)}\right) = ...
2
votes
0answers
73 views

Lyapunov function for a damped pendulum

The question is about damped pendulum. There are two statements I don't understand or I'm not sure if my justification for them is correct. Could you say if I'm right? The example is from a German ...
7
votes
2answers
56 views

Can $\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval containing $0$

If $p(x)$ and $q(x)$ are continuous functions for any $x$, can $y(x)=\sin(x^2)$ be solution of the diff equation $y''+p(x)y'+q(x)y=0$ in some interval $I=[a,b] $containing $0$? I think it is not as ...
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0answers
15 views

what happens when expansion parameter is of the order of dynamical variable itself?

Lets consider following differential equation, $\epsilon \frac{dy}{dt} = ....$ In principle one can use Method of matched asymptotic expansion or Method of multiple scales to solve such singular ...
4
votes
1answer
45 views

A question on ordinary differential inequality

Could we find a solution $f=f(x)$ to the following initial problem for the OD inequality? $$3xf'+f-\sqrt{6f}\leq 0,\quad f(0)=0,\quad f(8/3)=6.$$ . Added: The above question is in fact a special ...
0
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0answers
21 views

Trying to model a substance settling in water using an advection equation?

I am trying to model a substance dispersed in a container of water gradually settling at the bottom. I am considering only one dimension. The top is at $z = 1$, and the bottom is at $z = 0$. So at $t ...
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0answers
13 views

Perturbative expansion of eigenvalues

Consider the differential operator given by $L_{\epsilon}u := -u'' + \epsilon xu$ with $u(0) = u(\pi) = 0$. For $\epsilon = 0$, then the smallest eigenvalue of $L_0$ is $1$ with eigenfunction $\sin ...
0
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1answer
18 views

Is this correct solution of higher order differential equation?

Solve the equation $$\require{cancel} (D^4 + 4)y = 0.$$ Solution: The auxiliary equation is: $$D^4 + 1 = 0.$$ $$D^4 = -4.$$ $$(D^{\cancel{4}})^{\cancel{\frac{1}{4}}} = (-4)^{\frac{1}{4}}$$ $$D = ...