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
12 views

How to solve the following non-linear differential equation?

I'm having trouble solving the following differential equation: $y'(x)=\frac{8A^2}{(1+4A^2x^2)^2}\cdot y^2-4Bx$ $A$ and $B$ are real constants. I would be very grateful for any help. Thanks in ...
0
votes
1answer
12 views

Show $y_1(t) = y(t)\int^t_{t_0} \frac 1 {x_1(s)^2} e^{-\int_{t_0}^s p(r) dr} ds$ solves the 2-nd order ODE: $x'' + p(t)x' + q(t)x = 0$

Suppose $(I,y)$ solves the 2-nd order ODE: $x'' + p(t)x' + q(t)x = 0$. Assume $y(t) \neq 0$ for $t \in I$ and let $t_o \in I$. I want to show that $(I, y_1)$ where $$y_1(t) = y(t)\int^t_{t_0} \frac ...
2
votes
1answer
16 views

Eigenfunctions of the laplacian (1 dimension)

I have the following problem: $\frac{d^2 u}{dx^2}(x)+\lambda u(x)=0, x \in (a,b)$ and $u(a)=u(b)=0$. The general solution (for $\lambda>0$) is $u(x)=c_1\cos(\sqrt\lambda x)+c_2 \sin (\sqrt\lambda ...
0
votes
2answers
17 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
0
votes
0answers
5 views

How to find first-order quasi-linear PDEs form second-order quasi-linear PDE?

Transform $u_{tt} u_{xx}-u^{2}_{tx} + uu_{tt} + 1=0 $ into first-order quasi-linear PDEs. Attempt: $u_{tt}(u_{xx}+u)=(u_{tx}-1)(u_{tx}+1)$ To get $u_{tt} = u_{tx}-1\Rightarrow u_t = u_x ...
0
votes
0answers
18 views

Verify that $e^{at}$ is the only solution to the ODE: $y' = a y$ defined on $\mathbb R, a \neq 0$

Find all solutions to the ODE: $y' = a y$ defined on $\mathbb R, a \neq 0$ By inspection, I see that $e^{at}$ is a valid solution. However, my problem is to verify that $e^{at}$ is the only ...
0
votes
1answer
56 views

Why is -ln x is not equal to 1/ln x?

I am doing differential equation now and I need to convert them into the proper form in order to do my homogeneous differential equation. So now I just found out that -ln x is not equal to 1 / ln x. I ...
-2
votes
0answers
28 views

solve $y(x)+\int_{0}^{x}(x-s)y(s)ds=\frac{x^{3}}{6}$ [on hold]

Let $y:[0,\infty) \rightarrow \mathbb{R}$ be twice continuously differentiable and satisfy $$y(x)+\int_{0}^{x}(x-s)y(s)ds=\frac{x^{3}}{6}$$ then which of the following is true 1. ...
0
votes
1answer
69 views

Solving a first order linear ODE and determining the behavior of its solutions

(a) Draw a direction field for the given differential equation. How do solutions appear to behave as $t → 0$? Does the behavior depend on the choice of the initial value $a$? Let $a_{0}$ be the value ...
3
votes
0answers
29 views

Fredholm Integral Equations - Sturm-Lioville & Green Function Theory?

In an ODE's book one is given a 2nd order ode boundary value problem like $$y'' + A(x)y' + B(x)y = f(x), y(a) = y_a, y(b) = y_b$$ and might be told to analyze it with a Green function or via ...
1
vote
0answers
24 views

what is degree of given PDE…

When the given differential eqn is completely free from radicals the the final exponent on the highest order derivative amounts degree of given differential eqn. In present case it is 3 or 6. i.e. ...
0
votes
1answer
46 views

Set up differential equation

As people get older, they perceive time differently. The older one is, the faster time goes by. To quantify this issue, we create a model: The entire perceived period of time shall be $w(t)$ . A ...
0
votes
2answers
25 views

Mathematical Puzzle: A Drag Race of Who Wins

I'm having a real difficult time understanding how this problem is solved: "Two drivers, Alison and Kevin, are participating in a drag race. Beginning from a standing start, they each proceed with a ...
0
votes
0answers
18 views

Properties of the solution of an initial value problem

I have an IVP which can not be solved for explicitly of the form: $y''(t) = f(y)(y')^2 +ay'-g(y)$ $y(0)=0, y'(x_1) = h(y)>0$ with $y,y',y'' \in \mathbb{R}_+$, $x \in [0,x_1]$ and I know that the ...
1
vote
1answer
42 views

Given a solution of a differential equation, determine the differential eqution itself

Sorry if my layout is bad, I'm new. So this question was asked a couple of years ago on an exam about differential-equations. Suppose you have a third order differential-equation with the following ...
3
votes
2answers
34 views

How to solve differential equation $\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$

My task is to solve for $y$ from: $$\frac{d}{dx}\left(\frac{\lambda y'}{\sqrt{1+y'^2}}\right)=1$$ I have been given the answer, but I would like to calculate this myself also. $\lambda$ is a ...
0
votes
1answer
34 views

differential equation degree doubt

$dy/dx = sin^{-1} (y)$ this is a form of $dy/dx = f(y).$ so degree should be $1.$ but if i write it as $y = \sin(dy/dx).$ then degree is not defined as it is not a polynomial in $dy/dx $ please ...
0
votes
1answer
58 views

Checking: finding extremals for a functional

I'm trying to find the extremals of the functional $$J[y] = \int_0^1 (y')^2 + y^2 + 4ye^x \, {\rm d}x,$$ imposed that $y(0) = 0$ and $y(1) = 1 $. I got that there can't be extremals, and that's weird ...
0
votes
0answers
18 views

monotonicity of a $C^2(\mathbb{R})$ function

Let $c>0$ and $u(\xi)\in C^2(\mathbb{R})$ be a solution of $$ (D(u)u')'+cu'+g(u)=0,\qquad '=\frac{d}{d\xi} $$ with $c$. The assumptions for $D$ and $g$ are respectively $$D\in C([0,1])\cap ...
2
votes
1answer
24 views

the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$

Find all possible $(\lambda,u)$ where $\lambda \in \mathbb R$ and $u\ne0$, to the boundary value problem: $u''(x)+\lambda u(x)=0,x\in (0,1),$ $u(0)=u(1); u'(0)=u'(1).$ My Effort: for ...
0
votes
0answers
21 views

Differential equation in Maple : No solution on $x = -1 .. 1, y = -1 .. 1$.

Backround: Yesterday in class we had a lab session (practical work ?) on ODE and I have a question. We plot the following contour (I am using maple) implicitplot(H(x, y) = 0, x = -1 .. 1, y = -1 .. ...
1
vote
1answer
19 views

Cannot figure out a second order lineary differential equation with initial values

I got the following question: Solve the following initial value problem: $y(0) = 0$, $y'(0) = 1$, $$y'' + 10y' + 25y = 0$$ So I started with getting the general solution: $$ y(x) = C_1e^{-5x} + ...
0
votes
1answer
65 views

Solving a Linear ODE

Solve the following linear ODE $$3t^{2}y'+t^{3}y=cos(t)$$ What i tried Since this is a linear equation, i used the integrating factor method. First i didide both the LHS and RHS of the equation by ...
0
votes
0answers
20 views

Example: Solve a Second Order Nonhomogeneous ODE with Constant Coefficients by Variation of Parameters (2R-17)

Problem to solve: $$(D^2-2D+1)y=\frac{e^x}{x^3}$$ Answer in text: $$y=(c_1+c_2x)e^x+\frac12\frac{e^x}{x}$$ Our solution begins by rewriting the ODE in a more familiar form: ...
0
votes
1answer
34 views

Study of systems of Linear Differential Equations?

Is there any area of mathematics that deals with and formalizes systems of Linear DEs, akin to how Linear Algebra deals with systems of linear equations? Does it provide any insightful results?
0
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1answer
29 views

Linearity of a differential equation

The following is the general form of a linear ODE, where $t$ is the independent variable and $y$ is the dependent one: $a_n(t) \frac{d^ny(t)}{dt^n} + a_{n-1}(t) \frac{d^{n-1}y(t)}{dt^{n-1}} + \dots + ...
13
votes
0answers
3k views

Connection between the Laplace transform and generating functions

As I was sitting through a boring lecture rehashing basic techniques to solve ordinary differential equations, I began thinking about the Laplace transform and scribbled down a few ideas that I've ...
2
votes
0answers
30 views

Differential Equation Direction field

What i want to achieve: I want to plot the direction fields of the following three differential equations: 1. Malthusian growth model: $p'(t)=\lambda*p(t)$ with $\lambda=1$ and $p(t)=t$ 2. Linear ...
4
votes
0answers
34 views

How to show that a leaf is topologically a cone.

I am trying to understand the topological behaviour of foliations around irreducible singularities, specially in the case of singularities in the Poincaré domain. I am using the third chapter of this ...
0
votes
1answer
38 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
2
votes
1answer
37 views

Inverse of $3$ by $3$ matrix with non-constant entries.

I'm solving a question in nonhomogenous ordinary differential equation system $x'=Px+q$, and to solve my question I need to compute the inverse of the matrix $A=\begin{pmatrix}e^{-2t} & e^{-t} ...
1
vote
1answer
38 views

Why do we need sturm liouville form to solve ODE?

What is the reason that we have to recast a 2nd order ODE into SL form to find its eigenfunctions? for example, let $Ly=y''+y'+\frac{y}{4}=-ky$, boundary conditions $y=0$ at $x=0$ and $y-2y'=0$ at ...
0
votes
0answers
32 views

integral equation into differential equation

I have the equation $$ E = \alpha \int \int_S E dS $$ and I need to find a solution for E. My first instinct is to re-arrange it into a second order differential equation, but because dS is an area, ...
0
votes
1answer
29 views

Pass the lower limit to $-\infty$ for an integral of positive function

Hello I have an very elementary calculus problem. Let $\phi(\eta)$ be a real value function satisfying \begin{equation} \phi(-\infty)=1,\quad \phi(+\infty)=0, \end{equation} Let $g$ be a positive ...
0
votes
1answer
49 views

Solving a Variable Separable Differential Equation

The equation is $$y'=\frac{1}{18}x(81-y^2)$$ with $y(0)=81$, and I have to solve for an equation of the form $y(x)$ So I do $$\frac{dy}{(81-y^2)}=\frac{1}{18}x \ dx$$ I integrate both sides, and ...
1
vote
0answers
31 views

Solving a homogenous system of linear ODE with Pauli matrices

I was asked to solve find a general solution to $\overrightarrow{x'}=P\overrightarrow x$ where $P=\begin{pmatrix} -1 & 2 \\-1 & 1\end{pmatrix}$. Using the "regular" method of finding the ...
0
votes
0answers
17 views

Modulus of Green's function

Consider the nonlinear differential equation $$y'' = f(x,y,y')$$ together with the boundary conditions: $y(\alpha) = A$ and $y(\beta) = B$. Now $y(x)$ is a solution of this problem if and only if ...
0
votes
1answer
62 views

Find roots of $ω^x+(ω^x)^2+1=x$ [on hold]

We have to solve this equation at complex numbers group $ω^x+ω^{2x}+1=x$ I tried to find the roots, which led to $x = 0 , 3 $ But $0$ isn't right
0
votes
1answer
39 views

Solving an equation by Laplace transform

Consider the following equation: $$ y^{\prime\prime}(x) +x = \int _0 ^x (x-u)y(u)du \qquad y(0)=0 \quad y^{\prime}(0)=1$$ I solved it by Laplace transform and got $-\sinh x$ as a solution. It is ...
2
votes
1answer
21 views

Extracting differential equations [duplicate]

$$\frac{dx}{dy} = \frac{x(\alpha - \beta y)}{y(\delta x - \gamma)}$$ How do I extract two differential equations (y as a function of x and x as a function of y) from the equation above? I could ...
0
votes
2answers
50 views

Solving a first order ODE

Consider the initial value problem $$y'=ty(4-y)/(1+t)$$ $$y(0)=y_{0}>0$$ (a)Determine how the solution behaves as $t$ tends to infinity. (b)If $y_{0}=2$,find the time $T$ at which the solution ...
1
vote
0answers
21 views

Particular Solution of ODE 2?

today I found an interesting example considering viscoelasticity. While I was solving the given ODE I wondered, how the authors came to the solution they gave ... (the thing I am talking about is ...
0
votes
1answer
24 views

Second order differential equation, physics.

I need your input on this exercise I'm doing: "A 2-kg mass is suspended from a string. The displacement of the spring-mass equilibrium from the spring equilibrium is measured to be 50 cm. If the mass ...
0
votes
0answers
17 views

Find a Lyapunov function

How can I find a function of Lyapunov ? is there specific methods ? For exemple, how can I find the Lyapunov function of ...
-3
votes
2answers
37 views

Where is f and gnot differentiable? [on hold]

is my answer correct ???? I try to solve it but not sure if correct or not please help
0
votes
0answers
15 views

How can I solve this differential equation with fourier series?

Find a formal solution $u(x; y)$ by using Fourier series. (Hint: In two dimensions the basis functions have one of the forms $\sin(ax) \sin(by)$, $\sin(ax) \cos(by)$ and $\cos(ax) \cos(by)$, with ...
2
votes
0answers
5 views

Convergance of DASPK for a non-linear DAE

I have a system of non-linear DAE and I noticed that the system does not converge if some of the equations are not differentiated. For example, if the control volume equation is represented as this: ...
0
votes
1answer
30 views

Stationary function $y=y(x)$ of the integral $\int_0 ^4 (xy'-(y')^2)dx$ [on hold]

Find the Stationary function $y=y(x)$ of the integral $\int_0 ^4 (xy'-(y')^2)dx$ satisfying the condition $y(0)=0, y(4)=3.$
0
votes
1answer
50 views

How to solve $y'''-y=x+1$?

Solve ODE: $$y'''-y=x+1$$ To find the particular solution, I thought to impose $$y_p(x)=ax^3+bx^2+cx+d$$ Fair Enough? Or should I consider other?
0
votes
1answer
17 views

Solving first order discrete differential equation

I have a question about solving a first order discrete differential equation. The equation is $x' = Ax$ with $x_{0} = x0$ I found Runge Kutta could solve the differential equation, but required ...