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

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4
votes
5answers
74 views

Integrate $\int \left(A x^2+B x+c\right) \, dx$

I am asked to find the solution to the initial value problem: $$y'=\text{Ax}^2+\text{Bx}+c,$$ where $y(1)=1$, I get: $$\frac{A x^3}{3}+\frac{B x^2}{2}+c x+d$$ But the answer to this is: ...
0
votes
3answers
847 views

How does one solve $ y' = ( {2+\sqrt x})/({2+\sqrt y})$?

How does one solve $\displaystyle\frac{dy}{dx} = \frac{2+\sqrt x}{2+\sqrt y}$? I tried moving $x$ equation to left side and $y$ equation to the right side, but that results in ...
0
votes
0answers
28 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
1
vote
3answers
342 views

How to solve $y' = \sqrt {x+y+1}$

How does one solve $y' = \sqrt{x+y+1}$? I try substituting $v=x+y+1$ and using substitution methods, but it turned out to be so messy.
3
votes
0answers
29 views

Rolling parabola & catenary

By rolling a rigid catenary on a straight line one obtains the locus of its center of curvature as a parabola. This is well known as the natural equation connecting arc length and radius of curvature ...
1
vote
3answers
137 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
2
votes
4answers
47 views

Differential equation $(x-3)(\frac{dy}{dx})+y=6e^x, x>0$

I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set this up?
1
vote
3answers
78 views

Why does solving $\int \frac{v}{9.8-0.0025v^2}\mathrm{d}v=\int1{d}x$ for $v^2$ in terms of $x$ produce 2 completely different answers?

In this question $g=9.8$ (acceleration of free fall). You are also given that when $x=0$ $v=0$. My answer is $v^2=400g(1-e^\frac{x}{200})$. I obtained it by integrating both sides so that ...
0
votes
0answers
27 views

Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
1
vote
1answer
24 views

Discretization of an Euler-Bernoulli

Given the following Euler-Bernoulli equation: $$ (s(x) w(x)'')''= q(x),\ \ x \in [0,1]$$ Could someone explain why the following discretization scheme may not be a good idea? \begin{align*} ...
0
votes
0answers
18 views

Represent this differential equation as a set of n+1 equations w n+1 unknowns

Given the following differential equation: $$s''w'' + 2s'w''' + sw'''' = q$$ We use these approximations: $$w''''(x_i) \approx \frac { { w }_{ i+2 }-4{ w }_{ i+1 }+6{ w }_{ i }-4{ w }_{ i-1 }+{ w ...
4
votes
1answer
36 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
4
votes
1answer
71 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
4
votes
3answers
114 views

Solution of $\frac{d^2y}{dx^2} - \frac{H(x) y}{b} = H(-x)$

Does the equation $$\frac{d^2y}{dx^2} - \frac{H(x)}{b} y = c H(x)$$ have a solution where $H(x)$ is the Heaviside step function and $b$ and $c$ are constant? Update: What about the second step ...
2
votes
1answer
28 views

In what sense is the Ricci-Flow equation a “distant relative” of the Black-Scholes equation?

In the book "The Poincare Conjecture: In Search of the Shape of the Universe" by Donal O'Shea, the author states that, "The Ricci-flow equation Perelman wrote, a type of heat equation, is a distant ...
3
votes
1answer
27 views

Solution of differential equations with discontinuity

Suppose that we have scalar differential equation \begin{equation} \dot{x}(t)=u(t) \end{equation} Here $u(t)$ is a piecewise constant function with discontinuity. If the points of discontinuity is ...
1
vote
1answer
768 views

Legendre polynomials recurrence relation

How can i get? $$P_{n+1}=xP_n(x)-\frac{1-x^2}{n+1} P'_n(x)$$ $n>=0$ Also know as the leadder equation of the legendre polinomials i tried to use de recurrence relations as: ...
0
votes
1answer
27 views

Matrix Solution

I have matrix integral equation of the following form ${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ . All dimensions are indicated in equation itself. " ' " ...
7
votes
0answers
134 views

How to solve a time-dependent Schrodinger equation in periodic Dirac delta potential

I'm trying to solve a 1D time-dependent Schrodinger equation: $$ i\frac{\partial \psi(x,t)}{\partial t}=\left[-\frac{1}{2} \frac{\partial^2}{\partial x^2}+V(x)+F(t)*x\right]\psi(x,t) $$ where $V(x)$ ...
1
vote
0answers
85 views

Second order, inhomogeneous, linear differential equation

I come across this equation in book $$F(z)=(1-\lambda + \mu )f(z) + (\lambda - \mu) zf'(z) + \lambda\mu z^2f''(z)$$ where $\lambda \not= 0$ and $\mu \not= 0.$ My question is how to find $f$? Can ...
0
votes
0answers
23 views

Equation of vector field with rotation [closed]

Solve this equation of vector field F,c is constant. $$\nabla (\nabla \cdot F) = c\nabla \times F$$
0
votes
1answer
19 views

Book suggestion for practicing tough Ordinary DE problems

I am preparing myself for a post undergraduate (masters) entrance exam in mathematics. Can someone suggest a really good practice material with challenging questions of all types for ordinary ...
0
votes
1answer
35 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
1
vote
1answer
111 views

differential equations, diagonalizable matrix

I have a question of differential equations of the form. $\textbf{x}'(t)=A*\textbf{x(t)}$, where x is an n-dimensional matrix, and A is an n*n real matrix. I have learned to solve this if a is ...
1
vote
1answer
17 views

Convolution and Total Response Differential Equations

Convolution with differential equations is extremely confusing to me. The two following questions were asked in class and we were asked to think about them. I want to work them out but I don't know ...
1
vote
1answer
58 views

Second-order ODE with substitution

I’m struggling with this question: Use the substitution $y(t) = z(t)\,e^{-t}$ to transform the ordinary differential equation $$\frac{d^2 y}{dt^2} + 2\,\frac{dy}{dt} + y = t^2 e^{-t}$$ into an ...
0
votes
1answer
31 views

ODE with Laplace transform: the jump of $\dot y$

I solved this eq. using the Laplace Transform: $\ddot y+4\dot y+13 y=\delta(t-2\pi)-\delta(t-7\pi)$ The sol. is: $y(t)=\frac{1}{3} e^{2 t} (-e^{14 \pi} \theta(t-7\pi) sin(3 t)+e^{4 \pi} \theta(t-2 ...
9
votes
3answers
639 views

How do I solve $\vert x\vert^{x^2-2x} = 1$?

I have the exponential equation $\vert x\vert^{x^2-2x} = 1$, but how do I solve it?
-2
votes
0answers
27 views

Eulerian or Lagrangian [closed]

I have a partial differential equation, which the author who have solved this equation told me he solved it base on Lagrangian approach not Eulerian. I do want to know what are these two approaches ...
0
votes
1answer
59 views

Matrix-valued differential equation $A'(t)=A(t)B(t)$

How to solve matrix-valued differential equations of type $$A'(t)=A(t)B(t) \tag 1$$ All the given functions are square matrices of dimension $3$ and only $A(t)$ is invertible (not $B(t)$ or ...
1
vote
2answers
70 views

Solution to $\dfrac{dy}{dx} + \dfrac{y}{x}=e^{xy}\text{cos}^{2}x$

Good evening, may I know if my solution to the a/m problem is correct? Thank you. Let $v=xy. $ Then $\dfrac{dv}{dx}= y+\dfrac{dy}{dx}x=xe^v\text{cos}^2{x}$ Hence, $\begin{align}\int e^{-v} dv = \int ...
1
vote
0answers
61 views

Matrix exponent form

We have an equation of matrix exponent $ Ae^{Ax}R-e^{Ax}R (P_1 +P_2 x) = Y \tag1$ Given condition $A,R,P_1,P_2,Y$ are constant $3 \times 3 $ matrices. R is invertible,orthonormal,determinent ...
0
votes
0answers
46 views

Integral of $\exp(-x\,f(x))$

What is the evaluation of the integral of the following form or is there any alternative form for it? $$\int e^{-x \, f(x)} dx \tag 1$$
4
votes
0answers
49 views

Solution for $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$

I am looking for the function $f(x)$ that solves $\frac{a}{x} = \int_0^1 \frac{f(z)}{\left(f(x)+f(z)\right)^2} dz$ such that $f(0)=0$. Even hints how to approach to this question would be very ...
4
votes
2answers
40 views

Differential Equations with Discontinuous Forcing Functions

$$ y''+y'+1.25y = g(t), \quad t > 0, $$ $$y(0) = 0, \quad y'(0) = 0 $$ $$g(t) = \left\{ \begin{array}{ll} \sin{t} & 0 \le t < \pi \\ 0 & t \ge \pi \end{array}\right.$$ ...
1
vote
1answer
25 views

Diff. Eq. Example with Matrices

I'm currently working on a side project of mine that deals with $\sin(A)$ and $\cos(B)$, where $A,B\in\mathbb{C}^{nxn}$. I'm trying to find some interesting (or non-interesting) examples where one ...
3
votes
2answers
80 views

Symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$

I was trying to solve an exercise in one of Arnold's book that asks for the symmetry group of the vector field $V=x \partial /\partial x + y \partial /\partial y$, that is the diffeomorphisms $g$ of ...
3
votes
2answers
82 views

Solve ODE $(1+x^3)y'=x$

I was doing problems from Simmons and got stuck at this: $$(1+x^3)y'=x$$ If it was $x^2$ instead of $x$ then we could simply substitute $1+x^3$ for some variable, but that is not the case. I also ...
1
vote
4answers
64 views

How to approach the ODE $(y\cos y-\sin y+x)y'=y$?

I am not able to understand how to approach the following ODE. It is neither exact nor homogeneous nor perhaps linear. Please help \begin{aligned}(y\cos y-\sin y+x)y'=y\end{aligned} The given ...
0
votes
1answer
14 views

Cannot obtain a normalised eigenfunction for a boundary problem.

so I have done most of the problem, but I cannot solve the part where I have to do an integration. The problem says: "Determine the normalised eigenfunctions of the boundary problem $y''+\lambda ...
0
votes
1answer
37 views

What are the ordinary and singular points of the first order diff. equation?

Consider a first order differential equation. What do ordinary and singular points mean? What do they represent? (I cannot understand their formal definitions so please explain with examples. Thank ...
2
votes
3answers
63 views

Conversion of rotation matrix to quaternion

We use unit length Quaternion to represent rotations. Following is a general rotation matrix obtained ${\begin{bmatrix}m_{00} & m_{01}&m_{02} \\ m_{10} & m_{11}&m_{12}\\ m_{20} & ...
1
vote
2answers
33 views

Solving the following (fairly simple) differential equation…

I need to solve the following differential equation: $$y'\cos^2x+y=\tan(x)$$ I have tried to solve it using the integrating factor $e^{\int (1/\cos^2x) \mathrm{d}x}$, but things got messed up. How ...
2
votes
1answer
67 views

ODE $d^2y/dx^2 + y/a^2 = u(x)$

Does the following ODE: $$d^2y/dx^2 + y/a^2 = u(x)$$ have a solution? where $u(x)$ is the step function and a is constant.
3
votes
1answer
47 views

solve $\cos y \sin(2x) dx + (\cos^2y - \cos^2x)dy = 0$

any ideas on how to approach this ? Few observations : 1) not separable even after simplifying trig 2) not exact I am thinking substitution may work, still trying... appreciate any help :)
0
votes
1answer
79 views

Existence and uniqueness for initial value problem with $y'=1/(1+|y|)$

I have the following problem to solve and am not certain of whether my attempt at the solution is correct. Problem description: Given an IVP $\dot{y} = \frac{1}{1+|y|},y(0)=y_0\in\mathbb{R}\forall ...
3
votes
2answers
37 views

Solving the differential equation $\frac{dy}{dt} = \frac{t+1}{y+1}$

I am working on the differential equation $$\frac{dy}{dt} = \frac{t+1}{y+1}, \quad y(1)=2$$ Progress so far: $$(y+1) \, dy = (t+1)\,dt$$ $$\int y+1 \ dy = \int (t+1)\,dt$$ $$\frac{y^2}{2} + y = ...
5
votes
3answers
54 views

Solving the differential equation $\frac{dy}{dt}=e^{t-y}$

I am working on the equation $$\frac{dy}{dt}=e^{t-y},\qquad y(0) = 1$$ This is what I have tried to get it to its exact solution: $$\frac{dy}{dt}=e^{t}e^{-y}$$ $$\frac{1}{e^{-y}}dy=e^{t}dt$$ ...
0
votes
2answers
43 views

Probability density function that evolves with time according to a delay differential

Consider a real valued variable $X(t)$ that evolves with time according to the delay differential $\frac{dX(t)}{dt} = \alpha X(t-t_0) \int_{t_0}^\infty f(y) h(t-t_0,y) dy - \beta X(t) ...
0
votes
0answers
24 views

Is my function singular at these two points?

My function $S(x,y,t)$ satisfies the following PDE $$\frac{\partial S(x,y,t)}{\partial t}=-H(x,y)$$ where the known function $H=x^2+xy+y^2+\frac{1}{x-\alpha}+\frac{1}{\beta-x}$. It is clear that $H$ ...