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

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0
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2answers
27 views

Finding Explicit Form of Function Defined by Definite Integral

Let $$f(y) = \int_{-\infty}^{\infty} e^{-x^2} \cos (xy) \> dx$$ One can show that $$f'(y) = - \int_{-\infty}^{\infty}xe^{-x^2} \sin (xy) \> dx$$ I'm interested in making an ODE involving $...
1
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1answer
44 views

General solution of a nonlinear differential equation

Nonlinear differential equation gone beyond my field of expertise but I'd like to know the details of a problem and to do that I should know the general solution of the following nonlinear ...
10
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0answers
72 views

Solving $f'(x) = f(f(x))$

Is there any solution to the differential equation $f'(x) = f(f(x))$? I couldn't find any information on this kind of DE
3
votes
2answers
69 views

Should we re-define Sine?

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle. However I think these ...
-2
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0answers
18 views

The Good or The Use of Equations [on hold]

-Please, Can somebody help me by telling me what's the good or the use of the following equations in our daily life: Quadratic Equation. Cubic Equation. Linear Equation. Laplace's Equation. ...
0
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1answer
41 views

solution of the ODE $u du =ydx+xdy$

In this case $u=u(x,y)$. When I saw this I just went on to taking iindefinite integral both sides yielding $ u^2=4xy+K $. Yet, the book I am using now got $udu=d(xy)$, which yields $ u^2=2xy+K$. I'm I ...
3
votes
2answers
95 views

Solution of $f(x)^2\dfrac{d^2}{dx^2}f(x)=x$

I am stuck in finding the solution of this apparently simple differential equation: $$f(x)^2\dfrac{d^2}{dx^2}f(x)=x$$ with$f(0)=a$ and $f(0)'=b$ Using Maple the solution seems to be a combination of ...
0
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2answers
77 views

Difficult engineering second order DE, any pointers?

I have the following engineering DE: $$rR''+R'+\alpha r(R^2_0-r^2)\lambda^2R=0$$ Where $R(r)$ is Real, $r \geq 0$, $\alpha >0$. Boundary conditions $R(R_0)=0$ and $\Big(\frac{dR}{dr}\Big)_{r=0}=...
0
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3answers
57 views

How to integrate the following: $\int{\frac{2y'y}{y^2+1}dx}$

I have encountered the following problem: $\int{\frac{2y'y}{y^2+1}dx}$ According to wolfram the solution is: $log(y^2 + 1)$ How was this solution derived and which rules were used?
6
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4answers
1k views

How do you solve the following separable differential equation: y'y = y + 1?

I just started learning about differential equations and encountered following equation: $$ y'y = y +1 $$ Wolfram alpha provided the following explanation: here But I'm not sure how the integration ...
1
vote
1answer
32 views

Solving ODE for practice

I'm doing self study and I can't solve this equation: $$ax + \ln y = y + b$$ Where I'm supposed to eliminate the arbitrary constants. The given answer is $(y - y^2)(y'') = (y')^2$ But my workings ...
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0answers
16 views

Singularities in a PDE

This is more of a general question rather than anything specific but I was just wondering if someone could point me toward resources which discuss singularities in a PDE rather than in an ODE (by ...
1
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2answers
79 views

Nonlinear 2nd order ODE

I have been looking at numerical solutions to the following nonlinear Bessel-type ODE: $$ xy'' + 2 y' = y^2 - k^2, $$ where k is a constant. In general, $y = \pm k$ is an asymptotic solution, and as $...
0
votes
1answer
35 views

Easier solution to first order non-linear differential equation?

Im am dealing with this differential equation: $$m\frac{dv}{dt}=mg-kv^2$$ where $m,g,k$ are constants. I am able to solve this by treating this as a separable differential equation, but that method ...
0
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1answer
24 views

How to compute a Slope for a 3 or multi dimensional equation.

If I have an equation Z=X^2+Y^2+3X+6Y+5 and want to find the slope at the point x=2 , y=1 .How do we compute it ?I know for a two dimensional equation we can compute it by differentiation of Y with ...
2
votes
2answers
58 views

Sum of square of function

If $f'(x) = g(x)$ and $g'(x) = - f(x)$ for all real $x$ and $f(5) =2 =f'(5)$ then we have to find $f^2$$(10) + g^2(10)$ I tried but got stuck
0
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1answer
48 views

Is going from $V_{\text{L}} = L \frac{di_{\text{L}}}{dt}$ to $\frac{ V_{\text{L}} } {i_L} = L \frac{d}{dt}$ allowed?

The Laplace transform of $\frac{d}{dt} f(t)$ would be sF(s), when f(0)=0, which is something you can find in a Laplace transform table. If there is a rule that prohibits mathematical operations from $...
0
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0answers
30 views

Heat Equation : Commutation of partial derivatives and summation

I'am having a problem when checking the validity of the solution i found for the heat equation: \begin{cases} U_{t}(x,t)=U_{xx}(x,t),\ {(x,t)\in (0,1)\times(0,+\infty)} \\ U(x,0) = x^2 - x\\U(0,t)=0\...
1
vote
1answer
24 views

Implicit method for ODE

I want to numerically solve the initial value problem of ordinary differential equation for function $u=u(t)$: $$ u'(t)=L(u). $$ I find an second-order implicit method: $$ u_{n+1}=u_n+\Delta t L(u_{n+...
0
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2answers
22 views

Matrix Differentiation of $-a^T X^T y$ on $a.$

In short; what is the correct differentiation of: $$S(a)=-a^TX^Ty$$ when differentiating: $$0=\frac{∂S}{∂a}= \;?$$ Long story is; I know that: $$J(a)=\underbrace{\:\:\:a^TX^TXa\:\:\:}_u\:\...
2
votes
0answers
28 views

'2nd order' Picard Iteration

I'm self-studying differential equations using MIT's publicly available materials. One of the problem set exercises deals with what I'm calling a second order Picard Iteration. To be explicit, we ...
0
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0answers
11 views

Are there any examples of higher order ireducible linear differential operators?

Given a monic, linear differential operator $L = D^n + f_{n-1}(x)D^{n-1} + \dots + f_1(x) + f_0(x)$, say $f_0, \dots, f_{n-1}$ analytic for simplicity's sake, we say that $L$ is irreducible if there ...
1
vote
1answer
382 views

is there are specific way to solve coupled first-order differential equations with coefficients varying?

suppose I have "n" coupled differential equation represented by the matrix, Y• = A Y , where Y• is the column matrix containing first derivatives, namely, y1•(t), y2•(t), ... yn&...
3
votes
1answer
26 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
0
votes
1answer
31 views

Differentiation of$ f^{-1}(x)$, where $f(x)=e^{x-1}+x^3-4x^{-3}+10$

if $f(x)=e^{x-1}+x^3-4x^{-3}+10$ then find $\frac{d(f^{-1}(x))}{dx}$ at $x=8$..... (here $f^{-1}(x)$ means inverse of $f(x)$) I was trying to solve this problem but was not able to find out the way ....
0
votes
2answers
27 views

Solve the following IVP with explicit solution

Given: $4 dx + 2 {cos(y)\over sin(y)} dy = 0, \qquad y(0) = {\pi\over 2}$ I've already test the exactness which is $0$ for the result of both derivatives. Then I found the potential function is ...
0
votes
1answer
19 views

How to find the total derivative of a function $f_a(y(t),x(t))$ subjected to parametric change with the parameter $a$

It is well known to find the total derivative of a function $f(x(t),y(t))$. I consider it as $Td_f$. What, if the function depends upon some parameter, say, $a$. Then, how to find the total derivative ...
-1
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4answers
52 views

Finding the polynomial [on hold]

Find a nontrivial polynomial function $p(x)$ such that $p(2x)=p'(x)p''(x)\not=0$
2
votes
1answer
177 views

Numeric solution of third order ODE

I need to solve the following third order (non-linear) ODE by numerical methods: \begin{equation}\tag{1} h^{3} \dfrac{d^3 h}{d x^3} = h-1. \end{equation} By assumption, the solution should approach $ ...
8
votes
4answers
2k views

How unique is $e$?

Is the property of a function being its own derivative unique to $e^x$, or are there other functions with this property? My working for $e$ is that for any $y=a^x$, $ln(y)=x\ln a$, so $\frac{dy}{dx}=\...
3
votes
3answers
61 views

Second-order non-linear ODE

$2tx'-x=lnx'$ I differentiated both sides with respect to x: $x'+2tx''=\frac {x''}{x'}$ Substituting $p=x'$, $p+2tp'=\frac{p'}{p}$ But I have no clue what can I do from here on. EDIT: $t$ is the ...
1
vote
2answers
383 views

Problem related with boundary value problem and eigenvalue, eigenfunctions

I was looking at previous year exam papers and was stuck on the following problem: For the boundary value problem, $\,\,y''+\lambda y=0; y(0)=0,y(1)=0, \,\,\exists$ an eigenvalue $\lambda$ ...
1
vote
1answer
23 views

First principle of differentiation needs to approximate a sufficiently small integral as area?

$$y(t+\Delta t) = e^{- \int_{t}^{t+\Delta t}H(t')dt'}y(t)$$ is the solution to the differential equation $$\frac{dy}{dt} = -H(t)y$$, $H(t)$ and $y$ are scalar. However, in showing that $$y(t+\Delta t)...
0
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1answer
30 views

How small need it be to approximate integral as one area of product of initial value times length.

$$\left(\int_{t}^{t+\Delta t}a(t')dt'\right), a(t) \text{ is scalar}$$ How small need $\Delta t$ be to approximate $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right)$$ as $$a(t)\Delta t$$ [ Just one ...
2
votes
1answer
71 views

Integrating factor

Can anyone give me some hints as to how to solve the following question? I have to show that the equation below has an integrating factor of the form $t^2\theta^c$ where $c$ is an integer. $\...
0
votes
1answer
32 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
1
vote
0answers
43 views

Singularities in an Equation

these might sound like extremely trivial questions but since my background is more in probability and statistics I'm not too sure what to do, or even what to read up on to understand what to do. I ...
-4
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0answers
37 views

Help me solve this differential equation

I have came across where the complementary function has complex roots. Please help me solve this equation Also please use this method (operator d method ) as I know only this method I am uploading ...
0
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0answers
20 views

Development of Calculus and differential equation question

In my post Showing propagation in ordinary differential equation, I have a hard time getting the exponential solution of propagation$$y(t+\Delta t) = e^{- \int_{t}^{t+\Delta t}H(t')dt'}y(t)$$. I now ...
0
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0answers
43 views

Showing propagation in ordinary differential equation

I have a very simple linear First Order Homogeneous Differential Equation with time but this makes me ponder upon the definition of derivative and integral. $$\frac{dy}{dt} = -H(t)y$$ where $y, H(t)$ ...
0
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1answer
483 views

How to decide whether PDE is Homogeneous or non-homogeneous.

I am studying second order PDE. And I have seen homogeneous and non-homogeneous PDE. But I cannot decide which one is homogeneous or non-homogeneous. For examples; (1) $(D^3-3D^2D'+4D'^3)u=0$ ...
1
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3answers
56 views

Separation of Variables and Linear PDEs

Separation of variables is a powerful method which comes to our help for finding a closed form solution for a linear partial differential equation (PDE). For example, we all know that how the method ...
4
votes
4answers
110 views

How to solve $y''+y=x^2$?

I need to solve: $$y''+y=x^2$$ Taking the Laplace transform (and using the fact that it is a linear operator) on both sides I get: $$\mathscr{L}(y)=\frac{2}{s^3(s^2+1)}+y(0)\frac{s}{s^2+1}+y'(0)\...
0
votes
1answer
36 views

Time Derivative of a Positive Definite Matrix

Suppose we have a positive definite symmetric matrix $\mathbf V(0) \in \mathbb S^{n}_{++}$, which changes with time according to the following equation, $\dot{\mathbf V}(t) = \mathbf A \mathbf V(t) + ...
0
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0answers
41 views

mean evolution of 1D Fokker-Planck

Given the Fokker-Planck equation on 1D with drift term $D^{(1)}(x)$, and diffusion term $D^{(2)}(x)$, and the governing equation of probability density function $$\frac{\partial f(x,t)}{\partial t} = ...
0
votes
2answers
31 views

Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
0
votes
0answers
7 views

If we know $h_a = H_0^{1}(x + iy),$ for the Hankel function $H_0^{(1)}$, is it possible to determine $h_b = H_0^{1}(x - iy)?$

I am wondering if there is a certain identity for Hankel functions of the first kind of order $0$. If we know $$h_a = H_0^{(1)}(x + iy),$$ where $y > 0$, is it possible to determine $$h_b = H_0^{(...
0
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1answer
31 views
0
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0answers
19 views

Solutions to Strum-Louiville equation continuous even with discontinuous coefficients?

In the physics paper here (should be open access), the author first studies a Schrödinger equation in the form of a Strum-Louiville equation $$\frac{d}{dx}\frac{1}{m(x)}\frac{d}{dx}\phi(x) = -\...
0
votes
1answer
31 views

N-order differential equations

Suppose that we have n-order differential equation like $$h(x)=?$$ Is it possible to find a general solution for all n? $$(x^n+1).|h'(x)|^n=const.$$.