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

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1
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1answer
24 views

Solving ODE by substitution. Where does $dy$ goes

When solving ODE by substitution, where does $dy$ goes from the following example? $$\left(1+\frac{sin(y)}{cos(y)}\right)dy=x dx$$ Let $u=-cos(y)$. Hence $du = sin(y)$, which results in the following: ...
0
votes
1answer
35 views

Cauchy-Riemann equations on $f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}$

Let $$f(z)=\begin{cases}(z\overline{z}^{-1})^2&z\neq 0\\1&z=0\end{cases}.$$ I need to show that the Cauchy-Riemann equations hold for $f$ in $0$ but $f$ is not (complex) differentiable in $0$....
1
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1answer
42 views

Book tells me to solve separable diffeq with integrating factor, am I missing something?

Maybe this is for purely pedagogical purposes but the book I am using instructs me to solve $$ ydx+(1-x)dy=0 $$ By finding an appropriate integrating factor and solving. But $$ ydx+(1-x)dy=0\...
2
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2answers
63 views

Solving not exact differential equation $(x^2+y^2+y)dx-xdy=0$ (Riccati type?)

here is the diffeq: $$(x^2+y^2+y)dx-xdy=0$$ I only know how to solve non-exact equations like this when I can solve for a single variate integrating factor. I think this is a Riccati type equation, ...
0
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0answers
40 views

Real exponents on negative numbers

A textbook asks For which $p > 0$ is the solution of the IVP $$ \dot{x} = x^p, \quad x(0) = 1$$ unique and defined for all $t \leq 0$? This is simple for $p \geq 1$, but otherwise we will ...
0
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1answer
29 views

solve this simple differential equation

Just a small part of what i'm doing right now, I have to solve $d^2 f/ dx^2 =c f$ for a constant $c$ (which may be complex) and a function $f$ which might get complex values. the domain is $[0,1]$. ...
2
votes
1answer
34 views

Find the solutions to this ODE for an arbitrary $\lambda$

For $\lambda \in \mathbb{R}$, consider the boundary value problem $$x^2\frac{d^2y}{dx^2}+2x\frac{dy}{dx}+\lambda y=0, \quad x\in [1,2], \qquad y(1)=y(2)=0$$ Which of the following ...
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0answers
25 views

Special First-Order Differential Equation with Two Constraints

Let $D =[0,\overline{c}] \subset \mathbb{R}_+$ $\pi(c):D \to \mathbb{R}_+$ be continuously differentiable, bounded and strictly decreasing. $V(c):D \to \mathbb{R}$ be continuously differentiable, ...
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3answers
44 views

Solve the initial value problem for this inhomogeneous heat equation.

I'm trying to solve this IVP for heat equation, $$u_t-\frac{1}{4}u_{xx}=e^{-t}~~\text{ in }-\infty<x<\infty,~t>0,$$ $$u(x,0)=x^2.$$ By the superposition principle, the solution should equal ...
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2answers
90 views

How can it be proved that the limit definition and series definition of $e$ are equivalent, how do they model continuous growth?

The limit definition of $e$ gives the best intuition toward how $e$ can model continuous growth. $$\lim\limits_{n \to \infty} \left( 1 + \frac1n\right)^n$$ Let n represent n moments of growth that ...
2
votes
1answer
32 views

Laplace equation on a cylinder

For the Laplace equation in 3D $$\nabla^2 u =u_{xx}+u_{yy}+u_{zz}=0$$ in a right cylinder with an arbitrarily shaped base, whose top is $z=H$, bottom is $z=0$, we assume the following boundary ...
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0answers
23 views

Solving position vector/constants given the acceleration vector is a function of the velocity vector

Recently I've been working on a problem and it's consuming all my free time. I've cobbled together a small model that takes into account lift and drag and now I'm trying to fit it to real data. The ...
1
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2answers
61 views

Can Integration constant be anything?

When the question is to solve a given ODE (without initial value), can I assign any value to the constant $C$, in order to solve it, or is there a specific constant value that must be found? In ...
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0answers
23 views

intuition about system of coupled differential equations

I have a system with two components A and B where A is being depleted at a constant rate, and A activates B. Activation of B by A follows the Hill equation: $dA/dt = -r_{A}$ $dB/dt = V_m\...
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0answers
21 views

How do I solve this differential equations [closed]

show that : 1.$J_{1/2}(x)=\sqrt\frac{2}{\pi x}8x$ 2.$J_{1/2}(x)=\sqrt\frac{2}{\pi x}\cos x$ I don't know how to solve these problems .can someone help me out? please
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0answers
22 views

help with solution of linear differential equation [closed]

I want to solve the following problem using Runge-Kutta method: $\frac {dy}{dx}=xy$ for $x=1.4$ ,initially $x=1 , y=2$ (take $h=0.2$)
2
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0answers
31 views

solving differential equation [closed]

solve: $(1+p)^2r-2(1+p+q+pq)s+(1+p^2)t=0$ how do I solve this differential using monge's method? .Can someone help me with this
2
votes
1answer
28 views

Variation of parameters for ODES with distributions as coefficients

In my work, I encounter the following type of equation. Consider a non-homogeneous system \begin{equation} X'=A(t)X+f(t),\;\;\;\;\;\;\;\;(1) \end{equation} where $X$ is a $n$-dimensional vector valued ...
1
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1answer
56 views

Solution of differential equation $\frac{dy}{dx}=\frac{1}{xy(x^2 \sin y^2+1)}$ [duplicate]

Solve the given differential equation. $\frac{dy}{dx}=\frac{1}{xy(x^2 \sin y^2+1)}$ I have been trying to solve given differential equation using elementary approaches but no manipulation is a ...
1
vote
1answer
28 views

help with solving a simple differential equation [closed]

How do I solve this differential equation: $(x-y)dy=(x+y+1)dx$ This is simple but I am unable to solve this can someone help me with this?
4
votes
4answers
597 views

How do I solve $y''+4y=0$?

This problem is in Penney's Elementary Differential Equations, listed as a reducible, 2nd-order DE. The chapter has taught two techniques to be used, which are for when either $x$ or $y(x)$ is ...
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0answers
19 views

A system of two Riccati equations

I am trying to solve the following system of two non-linear Riccati-type differential equations: $$ \dot{x} = (1-x)\left(p^{x} + q^{x}x + qy\right), \\ \dot{y} = (1-y)\left(p^{y} + q^{y}y + qx\right)...
0
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1answer
48 views

how do I solve following differential equation [closed]

$\frac{dx}{z(x+y)}=\frac{dy}{z(x-y)}=\frac{dz}{x^2+y^2}$ How do I solve this equation I am not getting it can someone help?please
4
votes
1answer
88 views

Bounds on a system of coupled ODEs

Suppose we have a $1$-dimensional differential inequality $$\frac{dx}{dt} \leq x - x^3 $$ We can apply the Comparison principle to claim that if $y(t)$ is the solution to $\frac{dy}{dt} = y - y^3$, ...
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votes
1answer
43 views

how do I solve the following differntial equations [closed]

1.$(D^2+3D+2)y=e^{2x}8x$ and $\frac{d^2y}{dx^2}-4x\frac{dy}{dx}+(4x^2-1)y=-3^{x^2}8*2x$ I am not getting any idea how to begin solving these problems
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votes
0answers
58 views

Battle math equation [closed]

i am trying to code my own game and i am stuck at battle equation i will try to explain what i am trying to do as much as possible example of what i want to reach at the end http://2.bp.blogspot....
2
votes
1answer
61 views

Method for solving 2nd order linear PDE of three variables

For the 2nd order linear PDE below, please give method(s) to solve it, working, a solution, and what conditions the solution can exist? $$\sin(t)\frac{\partial^2y}{\partial t^2}+\cos(t)\frac{\partial ...
1
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2answers
48 views

Help with differential equation of unknown order

I was working with a problem and this differential equation came up. $$1+\frac {dx}{x}=e^{mx \, dy}$$ I don't think this has any closed form solution. Can anyone solve this if possible?
4
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0answers
54 views

How to integrate $\frac{x^n\,\,\,\sqrt{x+1} }{\left(m (x+1)^3-x\right)^{3/2}}$?

Let $$ f(x)=\frac{x^n\,\,\,\sqrt{x+1} }{\left(m (x+1)^3-x\right)^{3/2}} $$ $$ 0<m<\frac{4}{27}\,\,\,\,;\,\,\,\frac{1}{2}<n<1 $$ I am trying to integrate the above function. I tried ...
1
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0answers
33 views

problem regarding exponential solution of heat equation?

Let the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2}+ \frac{\partial^2 u}{\partial x_2^2}+ \frac{\partial^2 u}{\partial x_3^2}, \ t \geq 0 , \ x= (x_1, x_2, x_3)$...
1
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2answers
28 views

Question regarding differentiability of a composite function

I am given that the function $f:\mathbb{R}→\mathbb{R}$ is differentiable and am tasked to show that $A: \mathbb{R^n}→\mathbb{R^n}$ with $A(x):=f(\vert\vert x\vert\vert_2)x$ is also differentiable. At ...
-1
votes
2answers
37 views

Method of reduction of order: $x^2y′′− xy′+y=4x+3\ln(x)$

Looking to solve the following equation with method of reduction of order: $$x^2y′′− xy′+y=4x+3\ln(x)$$ I need to make the subsitution $x=e^t$
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3answers
43 views

Solving the IVP $8y''+26y=0$, $y(0)=2$, and $y'(0)=7$

Im supposed to find $y$ as a function of $t$ given the equation: $$8y''+26y=0$$ Initial values are $y(0)=2$ and $y'(0)=7$. I found the function of $y$ is $y=c_1\cos (\frac{\sqrt{13}}{2}t)+c_2\sin(...
0
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0answers
17 views

Operator Splitting, Piecewise Function, ODE,PDE,Matlab code [duplicate]

$t_f=1$ $$y(t) = \begin{cases} e^{2t},& 0 \le t \le t_f/2 \\ 2\left(t-\frac{t_f}{2}\right)+e^{t_f} , & t_f/2 \le t \le t_f \end{cases}$$ How can i write this piecewise function in ...
0
votes
1answer
26 views

Orbit direction in Hamiltonian systems

The system $$ \begin{pmatrix} \dot{x}\\ \dot{y} \end{pmatrix} = \begin{pmatrix} y\\ x(1 + 2x^2) \end{pmatrix} =: f(x, y) $$ has the Hamiltonian $$ H(x, y) = \frac{y^2 - x^2 - x^4}{2} $$ The orbits ...
2
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1answer
26 views

Brownian noise perturbing a differential equation

The following one-degree-of-freedom oscillator is given; $$\ddot{x}+kx=w(t),$$ where, $k>0$ and $w(.)$ is a Brownian noise perturbing the system. Assume we want to study boundedness of the ...
4
votes
3answers
6k views

Linear independence of function vectors and Wronskians

I am taking a course in ODE, and I got a homework question in which I am required to: Calculate the Wronskians of two function vectors (specifically $(t, 1)$ and $(t^{2}, 2t)$). Determine in what ...
0
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1answer
33 views

Finding all the eigenvalues and eigenfunctions for a BVP with an inequality condition

I am trying to find all the eigenvalues and eigenfunctions for the following boundary value problem \begin{eqnarray} \phi''(z) + \phi'(z) + \lambda \phi(z) &=& 0\\ \phi (0)&=& 0 \\ |\...
4
votes
1answer
54 views

Solving the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform, without missing solutions

I'm supposed to solve the ODE $y^{\prime\prime}(x)-y(x)=g(x)$ using the Fourier transform and then explain if I got the most general solution. First of all, I don't know what "solve" means here ...
1
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1answer
31 views

Frobenius method solution of this nasty 2nd order Linear ODE

I've tried but can't get the solution of this ode by Frobenius method. $(x^2)y''-6y=0$ I tried with $y=\sum_{k=0}^{\infty}(a_k \cdot x^{(k+r)})$ where $a_k$ is coefficient. I can't find the ...
1
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1answer
56 views

Solving coupled second order ODEs via Laplace transforms & Function theory.

I have used Laplace transforms to transform a system of 2 coupled second order ODEs into 2 simultaneous equations. 1st ode: $$\frac{3d^2y}{dt^2}+\frac{dy}{dx}=0$$ 2nd ode: $$\frac{5d^2y}{dx^2}-\...
2
votes
2answers
168 views

Tricky differential equation is solveable? $\frac{dy}{dt}=\sqrt{f(t)-y}$

It would be great to solve this problem! But I think maybe not possible because of the square root... is there anything that can be done with this? I guess this is invalid right: $$\left(\dfrac{dy}{...
3
votes
1answer
94 views

I have a special solution for the Lane-Emden equation. Can I use it to find the general solution?

The general Lane-Emden equation is $$\ddot{y}+\frac{2\dot{y}}{x}+y^N=0$$ where $y(0)=1$ and $\dot{y}(0)=0$. If we eliminate the requirement that $y(0)=1$ there is a special solution for all real ...
4
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2answers
157 views

Solve $f ' (x) + f '' (x)/2 = \sqrt f(x)$

How to solve the differential equation $$f ' (x) + f '' (x)/2 = \sqrt {f(x)}$$ Edit My efforts Assume $f(x) = a x^2 + b x + c$. Then we plug this into the differential equation $2 a x + b + a =...
1
vote
1answer
36 views

Dirichlet problem's solution on $[0,1]$.

The Dirchlet problem on $[0,1]$ is $$\begin{cases}{\Delta f=g}&\text{on } & (0,1) \\f(0)=a & \text{and}& f(1)=b \end{cases}$$ Where $\Delta=\frac{d^2}{dt^2}$. The unique solution of ...
0
votes
1answer
47 views

Analytic Solutions to differential (heat) equation.

I've searched around, but I couldn't find anything too helpful on the subject, so here goes. I am trying to find (if possible) an analytical solution to the differential equation of the form: $$ \pm\...
3
votes
2answers
52 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
1
vote
1answer
59 views

A general version of Gronwall's inequality

For the following $$|u(t)|^p\le C_1 \int_0^t |u(s)|^p\,ds+C_2$$ using Gronwall inequality, we have $$|u(t)|^p\le C_2(1+C_1 te^{C_1 t})$$ Now, my question is, for $$|u(t)|^p\le K_1 \int_0^t(1+|u(s)|^2)...
0
votes
1answer
43 views

First order differential equation - split on delta function

I have a couple of first order differential equations whose solutions I would like to approximate numerically in my python app. MATLAB ODEs solvers have a built-in detection of events. I do not see ...
0
votes
2answers
35 views

1st Order Nonlinear ODE

Verify that $y = −x^2$ is a solution for the equation $y' = x^3+ 2y/x-y^2/x$. Find the general solution for the equation. How would I go about solving this question?