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

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2
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1answer
27 views

How can you derive this integrating factor?

Consider the differential equation $M(x,y)+N(x,y)\frac{dy}{dx}$. If an integrating factor takes the form $\mu=\mu(xy)$, show that the necessary condition is $$\frac{N_x-M_y}{xM-yN}=F(xy)$$ I am not ...
2
votes
0answers
64 views

Please Help Me with Questions in Evans' PDE

I have stuck by the following two problems in Evans' PDE books (2nd edition), The problems require me to provide two counterexamples. problem 1: (see Page 307 Problem 12 ) U is open and $V \subset ...
0
votes
1answer
35 views

Is there analytic solution for my Eq

Is there analytic solution to this Eq : $$\frac{dy}{dx}=\sqrt{B(\frac{1}{y^2}-4y+3y^2)+Dy^2}$$ where $B,D$ are constants
2
votes
1answer
51 views

Finding all solutions to an initial value problem

This comes from a real analysis class, and I currently cannot assume any knowledge about integration. I want to find all solutions to the initial value problem $y' = y^{\frac{1}{2}}$, $y(0) = 0$. I ...
1
vote
1answer
58 views

Non homogeneous differential equations

What is the differential equation for $$ y=c_1e^{2x}\sin x + c_4 e^{2x} \cos x - xe^{-x}. $$ I'm not sure how to incorporate the last part of the solution into solving this problem?
3
votes
2answers
75 views

Find all functions such that $f'(t) = f(t) + \int_0^1 f(\tau)\,d\tau.$

From Spivak Find all functions such that $f'(t) = f(t) + \int_0^1 f(\tau)\,d\tau.$ My approach: differentiate both sides to get $f''(t) = f'(t)$, giving $f'(t) = Ce^t$, implying $f(t) = Ce^t + ...
0
votes
1answer
59 views

MATLAB Finding Difference of two equations

So I'm trying to figure out how to do a fairly simply mathematical task in MATLAB but I don't know what it is called so I don't know what to search. Basically what I want to do is if you have two ...
1
vote
0answers
35 views

Is my way of proving that y is even correct?

Let $x\in \mathbb{R}$ and $y:\mathbb{R} \rightarrow \mathbb{R}^n$. Let $F$ be a linear tranformation such that: $$F(y,x) = \frac{dy}{dx}$$ Assume $F(y(x),-x)=-F(y(x),x)$. Show that $y(x)=y(-x)$. ...
0
votes
4answers
135 views

Newtonian Mechanics - Differential equation

If we combine Newton's second law of motion i.e. $F=m\ddot{x}$ and Newton's law of gravity i.e., $$ F=G\frac{mM}{x^2}, $$ where $x$ is distance, we obtain the following equation: ...
0
votes
1answer
46 views

Finding a differential equation from $y = (c_1+c_2x)e^x+c_3$

I want to eliminate $c_1$, $c_2$ and $c_3$ from $y$. This is what I've done: $$y = (c_1+c_2x)e^x+c_3$$ $$y' = (c_1 + c_2(x+1))e^x$$ $$y'' = (c_1 + c_2(x+2))e^x$$ $$y''' = (c_1+c_2(x+3))e^x$$ $$y''' - ...
2
votes
1answer
81 views

Intuition behind the weight function

The inner product in a $L^2$ space can be defined as: $$\langle f,g\rangle =\int_a^b \bar{f}(x)g(x)w(x)dx$$ For Legendre polynomials, we define it as: $$\langle P_m,P_n\rangle =\int_0^1 ...
1
vote
3answers
82 views

How do I solve $\int \frac{1}{v(1+v^2)}dv$?

I'm trying to solve $y'=\frac{2xy}{x^2-y^2}$ and I ended up with the integral $\int \frac{1}{v(1+v^2)}dv$ as part of the solution. I got a big ugly solution using integration by parts, but I'm hoping ...
2
votes
0answers
215 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...
0
votes
0answers
53 views

Sturm-Liouville boundary value problem with two different eigenfunctions

I am trying to express a function $f(x)$ in terms of a complete set of eigenfunctions found from a Sturm-Liouville boundary value problem: $$2y''(x)+4y'(x)+\lambda y(x)=0$$ $$y(0)=0, y'(2)=0$$ For ...
3
votes
2answers
80 views

Justifying an ODE's solution

In an introductory lesson into ODEs, in order to "semi-rigorously" justify the solution for e.g. : $(a)\ \ y'+y=0$ we proceed without an ansatz or guess solution (hence the "semi-rigour"): Let: ...
4
votes
2answers
47 views

Where to look for “standard” ODE solutions?

I remember one day having stumbled upon a nice online resource where one could look for solutions to very general ODEs (or at least the literature names thereof), but unfortunately I forgot its ...
2
votes
2answers
85 views

Where have I gone wrong in trying to solve this ODE?

I'm trying to solve: $\frac{dy}{dx}=\frac{x+y-1}{x+4y+2}$. Attached is a picture of my working. Could someone please tell me where I'm going wrong? I'm tried both Maple and Wolfram and neither of ...
0
votes
1answer
666 views

Laplace equation in 1D with MATLAB - Dirichlet boundary condition

Here is a Matlab code to solve Laplace 's equation in 1D with Dirichlet's boundary condition u(0)=u(1)=0 using finite difference method ...
0
votes
1answer
62 views

Behavior of Lorenz Attractor

I'm trying to understand the behavior of the Lorenz attractor. I understand that it is very sensitive to initial conditions but I don't understand WHAT I'm looking at. So here are two lorenz attractor ...
1
vote
1answer
48 views

Population Dynamics Part 1

So I previously posted a long question consisting of 5 parts but I was advised to break it up into 5 questions. So the first part asks: If $A$ is a constant matrix, then $$ \vec{N}(t+\Delta t)= ...
1
vote
0answers
527 views

How can you find the Wronskian of Bessel Functions?

This is a homework problem I am trying to solve but I'm not sure if I'm doing it correctly, because it seems deceptively simple. Let $\alpha$ be a non-negative real constant. The differential ...
1
vote
1answer
66 views

Help Determine the equilibrium points and bifurcation value(s) for this family of DE.

Consider $x' = -x^4 + 5ax^2 - 4a^2$ a) Determine the equilibrium points and bifurcation value(s) for this family of DE. First I let $y = x^2$ Then set $-y^2 + 5ay - 4a^2 = 0 $ >>> $\frac{-5a +- ...
4
votes
1answer
124 views

Asymptotic estimate of an oscillatory differential equation

Let $f\in C^1(\mathbb{R} ,\mathbb{R} )$ and satisfying the condition: $$ \forall x >0, \quad f(x)>0, \forall x<0 , \quad f(x)<0 $$ Let $(\alpha, \beta) \in \mathbb{R^2}$. ...
0
votes
0answers
32 views

The integral $\int_A^B{FdR}$ is independent from the path $\Leftrightarrow$ $F=\bigtriangledown f$

Let $F$ be a vector field, $F=M\hat{i}+N\hat{j}+P\hat{k}$, where $M,N,P$:continuous at a region $D$. The integral $\int_A^B{FdR}$ is independent from the path from $A$ to $B$ in $D$ $\Leftrightarrow$ ...
1
vote
0answers
36 views

Stability and Asymptotic Stability of Rational Matrix Solutions

If $X(t)$ is a fundamental matrix solution of $\dot{x}=A(t)x$ on $a<t<\infty$ and suppose the entries of $X(t)$ are rational functions of the variable t in the form $x_{ij}=p_{ij}(t)/q_{ij}(t)$. ...
2
votes
3answers
122 views

Uniform convergence of matrix integral sequence

I was given recursively defined: $$ M_k(t)=I+\int_{t_0}^tA(s)M_{k-1}(s)~ds $$ and $M_0=I$ and that $A(t)$ is a matrix with entries that are continuous functions on $t_0\leq t\leq t_1$. By induction ...
0
votes
1answer
32 views

Solve autonomous equation

$ y''' - y = e^{2t} $ Actually I need only homogeneous solution of $ y''' - y = 0. $ I can clearly see that it is $ e^x, $ but how to prove it? I tried with $ \frac{dy}{dt} = v, $ but could ...
0
votes
2answers
43 views

Ordinary differential equations with signed first derivative

Consider the following coupled set of ordinary diferential equations: \begin{align} (K_{pa}+K_r)y_1(t)-K_ry_2(t)+C_0\operatorname{sign}(\dot{y}_1(t))\lvert\dot{y}_1(t)\rvert^\alpha &= ...
2
votes
1answer
73 views

Validating a PDE problem solution

I have the following problem, which I have tried to solve myself and I would like someone to verify that my answer is valid. The problem is the following: By separation of variables, derive the ...
1
vote
0answers
129 views

Inverse Laplace Transform without tables

How can I show that the Inverse Laplace transform of $\dfrac{a}{s^2+a^2}=\sin at$ without the use of tables? Is there a formula or a direct way to do this?
0
votes
1answer
62 views

Determine the interval in which the solution is defined

Find the solution of the following initial value problem or general solution in explicit form.Determine(at least approximately) the interal in which the solution is defined. $$y' \ = \ ...
2
votes
1answer
196 views

Use of Routh-Hurwitz if you have the eigenvalues?

This is for self-study of N-dimensional system of linear homogeneous ordinary differential equations of the form: $$ \mathbf{\dot{x}}=A\mathbf{x} $$ where A is the coefficient matrix of the system. ...
1
vote
1answer
96 views

Recursive coefficients for an infinte series in complex analysis / differential equations

I have a question that feels rather simple, but I seem to be stumped! Given that $f$ is entire, use a power series representation of $f$ about $0$ to solve the differential equation ...
0
votes
2answers
67 views

Inverse of $r sin(\omega t) + v t$?

I am wondering if there is an inverse for this function, $x(t)=r sin(\omega t) + v t$. The inverse function theorem suggests that an inverse for this function does exist, although it may have to be ...
2
votes
2answers
59 views

$\frac{\partial f(x,y)}{\partial x}=0$ whenever $f(x,y)=0\implies f(x,y)=g(y)$?

I'm reading a book on differential equations using symmetries, and at some point the author seems to imply that if $\frac{\partial f(x,y)}{\partial x}=0$ whenever $f(x,y)=0$, but $\frac{\partial ...
1
vote
1answer
91 views

How can solve this differential equation (fourth equation )?

How can I solve this differential equation? $$ \frac{dy}{dx}=\frac{1}{(\frac{1}{y}+\frac{3F}{y^2})}\sqrt{\frac{A}{y}+\frac{B}{y^2}+\frac{C}{y^4}+\frac{D}{y^5}+E} $$ where $A,B,C,D,E,F$ are ...
1
vote
1answer
71 views

How do I solve $y'=\frac{y}{x}\frac{x-y}{x+y}$?

I have a solution for $y'=\frac{x-y}{x+y}$ Now I have this new problem: $y'=\frac{y}{x}\frac{x-y}{x+y}$ My first approach was to solve it by parts; that way I could have re-used my solution for ...
0
votes
1answer
69 views

please solve this diffrential equation question on power series

In the differential equation $y'' + (x-3)y' + y=0 $ of power series at $x_0=2$ , I took $ y=\sum_{n=0}^{\infty}a_n(x-x_0)^n $ ,then I tried to solve this but not getting the answer. if someone solve ...
0
votes
1answer
31 views

Differentiable cauchy riemann equation

If f is differentiable and |f(z) = 7| in D(0,5) then f(z) is a constant function on this disk D(0,5). Is this true?
1
vote
2answers
75 views

Eigenvalues and eigenvectors of an integral operator

We have the following integral operator $$ Ku(t)=\int_0^1 G(t,s)\, u(s)\, ds,\,\, u\in L^2[0,1], $$ where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq ...
0
votes
1answer
37 views

How to separate $x^2X''v+xX'v+Xv_{yy}+x^2Xv_{zz} = 0$

I have the following partial differential equation: $$x^2X''v+xX'v+Xv_{yy}+x^2Xv_{zz} = 0,$$ where $X = X(x)$ and $v = v(y,z)$. How to separate it? Thank you for any help! From the book: ...
0
votes
2answers
64 views

ordinary differential equation equals to non zero integers

$$f ' '(x)+4f '(x)+f(x)=3\qquad f(0)=2\qquad f'(0)=1$$ Our teacher only taught us how to solve ODE equation $= 0$. which $α^2+4α+1=0$ but in this case I cant write like this since the equation $= ...
0
votes
1answer
52 views

Second Order Differential Equation Question

Got this question on my FP3 homework - if anyone could help me out I'd really appreciate it. .
0
votes
1answer
51 views

Separation of variables with three independent variables

I have the following differential equations problem: Derive sets of three ordinary differential equations from the following partial differential equation by separation of variables: ...
0
votes
1answer
50 views

Finding a general solution of a differential equation using the method of undetermined coefficients

I am to find the general solution of the differential equation $$y'' + y' + 4y = 4 \sinh(t) = 2e^t - 2e^{-t}$$ Now, using the method of undetermined coefficients, it is simple to arrive at the ...
1
vote
1answer
55 views

Understanding partial differential equation requirement

I'm reading about separation of variables in my Fourier series book and there is one requirement in a problem I don't understand. Here it is: I don't understand, why can't the $\sqrt{-A}$ take a ...
1
vote
1answer
52 views

Behavior of a dynamical system

I need some help understanding the behavior of a dynamical system. Here is the problem: Problem: Let $A$ be a square matrix of size $2$ with eigenvalues $\lambda=a \pm ib$ $(b \neq 0)$. I know that ...
0
votes
2answers
43 views

Prove that trajectory that starts in span of eigenvector will remain there

Assume we have a 2-d system of homogeneous ordinary differential equations:$$ \dot{\mathbf{x}}=\left[ \begin{array}{ c c } a & b \\ c & d \end{array} ...
0
votes
1answer
48 views

Find the solution $\Phi$ of an IVP

Find the solution $\Phi$ of the system \begin{align*} y_1'&=-y_1\\ y_2'&=y_1+ty_2\\ \end{align*} satisfying the initial condition $\Phi(0)=(2,1)$. It's been years since I've taken ODE, so I ...
2
votes
1answer
115 views

Use Fourier's method of separation of variables to solve the boundary value problem

Use Fourier's method of separation of variables to solve the boundary value problem comprising the following PDE and BC: PDE: $x \sin(y) u_x + \cos(y) u_y = -2 \sin(y) u $, $u = u(x,y)$ Boundary ...