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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0answers
13 views

RK4 for 2nd order ODE: multiplication by $h$

Take a first-order ODE $y' = f(x,y)$ for instance, with $y_0 = y(x_0)$. To calculate $y_1$ with RK4 one must first calculate $k_1 = f(x_0,y_0)$, and so on. Now look at the accepted answer (and ...
2
votes
1answer
35 views

Solving a differential equation: $y^4 \left(\frac{dy}{dx}\right)^4 = (y^2-1)^2$

How would one go ahead solving the following for these conditions: (i) passes through $(0,\frac{\sqrt{3}}{2})$ and (ii) $(0,\frac{\sqrt{5}}{2})$ : $$y^4 \left(\frac{dy}{dx}\right)^4 = (y^2-1)^2$$ ...
-3
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0answers
35 views

Solve $(x^2+2x)\frac{d^2y}{dx^2} -2(x+1)\frac{dy}{dx} +2y=0$ by series and elementary method [on hold]

Solve the following differential equation by series and elementary method: $$(x^2+2x)\frac{d^2y}{dx^2} -2(x+1)\frac{dy}{dx} +2y=0$$
0
votes
1answer
48 views

Differential equation $\frac{dy}{dx}=f(y),y(0)=y(1)=0$ [on hold]

Differential equation $$\frac{dy}{dx}=f(y),y(0)=y(1)=0$$ where $f:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous. Then $1.$ $y(x)=0$ if and only if $x\in (0,1)$ $2.$ $y$ is bounded $3.$ ...
1
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0answers
11 views

Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
1
vote
1answer
18 views

Mean Value Property - Nonnegative Harmonic function

I want to prove that the mean value property $$u(\textbf{x}_0) = \frac{1}{\pi r^2} \int \int _{\left \{ \left | x_0-x \right < r| \right \}} u(\textbf{x})d\textbf{x}$$ for non-negative harmonic ...
2
votes
2answers
78 views

$f '' - (f ')^2 + f=0$; what is known about solutions?

I'm curious about solutions to the equation $$f''-(f')^2+f=0$$ on the whole real line, as well as solutions which are periodic. Any info about the obvious multivariable generalization would interest ...
1
vote
2answers
37 views

$25 w(w-1)y''+(14-15w)y'+y=0$ - Gauss's Hypergeometric equation

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ near the singular point $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form ...
1
vote
1answer
30 views

Solve the following symetrical differential equation

Recently I encountered a differential equation which is as follows: $\frac{d^3y}{dx^3} + x^3 \frac{d^2y}{dx^2} + 3x^2 \frac{dy}{dx} + 6xy + 6 = 0 $ I couldn't solve it because we were not taught how ...
0
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1answer
18 views

Free oscillations and natural frequencies question

I'm given a mass-and-spring system with a couple second-order differential equations describing the behavior of the system. Those are: $$x^{\prime \prime} +50x - \frac{25}{2}y = 0$$ and $$y^{\prime ...
0
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0answers
43 views

Differential equations - Hypergeometric function [duplicate]

I would like to solve the equation $(x^2-x-6)y''+(5+3x)y'+y=0$ at $x=3$. I think we have to solve this problem in considering the Gauss's hypergeometric equation on the form (*) ...
-1
votes
1answer
19 views

Motion of a pendulum equation in the George Simmons book on differential equations [on hold]

I just can't understand the transition between this two formulas, why $dt$ becomes $T/4$. Can anybody help me with that?enter image description here
0
votes
1answer
18 views

Can this differential equation be transformed into an hypergeometric equation?

$$(1+x^2)y'' -4xy' + 6y = 0 $$ Can this be transformed into an hypergeometric equation of the form $x(1-x)y'' + (c - (a + b + 1)x)y' -aby = 0$? I know that we can do the transform is the term before ...
0
votes
1answer
8 views

Real solutions of $C_1e^{jx}+C_2e^{\bar jx}, C_1,C_2\in \mathbb{C}, j=\frac{1}{2}+i\frac{\sqrt 3}{2}$

Why does $C_1e^{jx}+C_2e^{\bar jx}, C_1,C_2\in \mathbb{C}, j=\frac{1}{2}+i\frac{\sqrt 3}{2}$ has the following result: $$S_{H_2}(\mathbb{R})=C_1e^{-1/2x}\cos(\frac{\sqrt ...
0
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0answers
53 views

How to solve this differential equation

I am unable to solve this equation. Please give me the way, how can I solve this? $$ y'''\sin x - y'e^x\cos x = 2 $$
0
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1answer
26 views

How to find Integrating factor of this equation

Equation $$ (x^2 e^{-y/x} + y^2 ) dx -xy dy =0 $$ Not Exact Q) Further solution is possible ? means Integrating factor
1
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0answers
29 views

Heat problem with an internal source of heat for which the maximum principle doesn't hold.

Heat problem with an internal source of heat for which the maximum principle doesn't hold. The problem is the following and honestly I don't know how to solve it... $$u_{t}=u_{tt}+2(t+1)+x(1-x) , ...
1
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1answer
36 views

How to graphically represent $\ddot x$?

We know that given a differential equation: $$\dot x = f(x), x \in X$$ The $\dot x$ is understood as the tangent vector on the solution trajectory $x$ lying in the tangent space of $X$ What about ...
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0answers
34 views

Find the solution of differential equation

This is a differential equation. I tried it many times but cant solve . xy'+(1+x)y=e^{-x}*sin2x My solution After finding ...
0
votes
2answers
41 views

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series

Solve $y'=e^{x^2}y$ (with $3$ terms only) in using power series. I know that $e^{x^2}=\sum_{n \geq0} \frac{x^{2n}}{n!}$, but I don't know how to find the coefficients $a_n$ in considering ...
0
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0answers
21 views

Spatial derivative of pendulum to time derivative

$m\ddot{x}\dot{x}+\frac{dV}{dx}\dot{x}=0\Rightarrow \frac{d}{dt}\left [ \frac{1}{2}m\dot{x}^{2}+V\left ( x \right ) \right ]=0$ $Using \frac{dV\left ( x\left ( t \right ) \right ...
2
votes
1answer
38 views

How to solve a differential equation

How would you solve this differential equation? I have been trying to use the technique of separable equations, but haven't got very far. $$\frac{D'(x)}{D(x)}=\frac{x}{1-x}$$
0
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2answers
10 views

Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$

Suppose that $F(s)=\mathcal{L}\{f(t)\}$ (Defined as the usual Laplacian operator). Show that if $c$ is a positive constant, then $\frac{1}{c}F(\frac{s}{c})=\mathcal{L}\{f(ct)\}$. ...
0
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0answers
18 views

Prove the following of the bessel series $xJ'_v(x)=vJ_v(x)-xJ_{v+1}(x)$

I can't figure out why the index of summation was incremented by one in the step indicated by the red line. Was this a misprint? If so, the proof would fail... proof
1
vote
1answer
32 views

Frobenius Method to Solve a Differential Equation

Having the equation $$x^{2}y''+xy'+x^{2}y=0$$ I get the indicial equation at get r=0, and am left with the equation. ...
1
vote
1answer
21 views

Finding the coefficient of the particular solution of $2y'(x)-3y(x)=cos(2x)$

Having to solve $2y'(x)-3y(x)=cos(2x)$ We have to find a method such that $y_2(x)=acos(2x)+b\sin(2x)$ the given result is $a=3/25, b=-4/25$, why the hell did I found result absolutely different: I ...
0
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0answers
8 views

Permute partial derivatives, ODE

Show that in the one dimensional case, we have $ \frac{\partial \phi}{\partial x} (t,x)= exp(\int_{t_0}^t \frac{\partial f}{\partial x} (s, \phi(s,x))ds)$, where $\phi(t,x)$ is the solution of the ODE ...
0
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0answers
11 views

Prove that there exists only one tangent solution

Let f be differentiable function.Prove that if function $x=tc_0 $ satisfies equation $x^{'}=f(\frac{x}{t})$ and $f^{'}(c_0)<1$ then every other solution of this equation is not tangent to ...
0
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1answer
22 views

Checking solution of $\frac1x \frac{∂}{∂x}[x\frac{∂f(x,y)}{∂x}]+\frac{1}{x^2}\frac{∂^2f(x,y)}{∂y^2}-c^2f(x,y)=0$

Here's what I did: Separating variables $$f(x,y)=X(x)Y(y)$$ $$\frac1x \frac{∂}{∂x}[x\frac{∂XY}{∂x}]+\frac{1}{x^2}\frac{∂^2XY}{∂y^2}-c^2XY=0$$ $$\frac{Y}x ...
3
votes
1answer
60 views

Differential Equation $\frac{dy}{dx} =\frac{x}{(x^2y + y^3}$

Please help with: $$\frac{dy}{dx} =\frac{x}{x^2y + y^3}$$ The hint says let $u = x^2$ I have tried all the possible substitutions and manipulations on this ODE and I just can't separate variables or ...
0
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1answer
25 views

linear ODE problem

A substance evaporates at a rate proportional to the exposed surface. If a spherical mothball of radius $\frac{1}{2}$ cm has radius $0.4$ cm after $6$ months, how long will it take: For ...
0
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1answer
12 views

How to find the differential of a series? And use it as a Substitution to solve the heat equation?

I have a question that says to solve the heat equation by substituting in $$\phi(x,t) = \frac{a_0(t)}{2} + \sum_{n=1}^{\infty} a_n(t)\cos(\frac{n\pi}{L} x)$$ I presume I must take the partial ...
0
votes
1answer
21 views

Numerical solution of ordinary differential equations, multistep method

I try to solve the following question, but I have no clue why we have $x'$ in the RHS: The formula $ x_{n+1} = (1-A)x_n + A{x_{n-1}} + \frac{h}{12}[(5-A)x'_{n+1}+8(1+A)x'_n + (5A-1)x'_{n-1}] $ is ...
2
votes
3answers
27 views

Need help with a quasi-linear PDE

Let be the following quasi-linear PDE: $(xu+y)\frac{\partial u}{\partial x} + (x+uy)\frac{\partial u}{\partial y} = 1 - u^2$. I wrote the characterstic system : $\frac{dx}{xu+y} = \frac{dy}{x+uy} = ...
0
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1answer
16 views

How to find λ values from a basis solutions

I try this question many time but i stuck on this how to find λ values in the below question. Find the differential Equation for which the given function form a basis solutions. ...
2
votes
4answers
49 views

ODE $y(1+2xy)dx+x(1-xy)dy=0$

$$y(1+2xy)dx+x(1-xy)dy=0$$ I have tried to isolate $\frac{dy}{dx}$ and got the following: $$\frac{dy}{dx}=-\frac{y(1+2xy)}{x(1-xy)}$$ but I understand that the terms have to be in the same ...
0
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1answer
21 views

Any help for solving this system of partial derivatives

Let the following equation : $(x-z)\frac{∂u}{∂x} + (y-z)\frac{∂u}{∂y} + 2z\frac{∂u}{∂z} = 0$. We write the characteristic system(to reduce it to an ODE system) so : $\frac{dx}{x-z}$ = ...
1
vote
1answer
19 views

Conversion to bessel equation

I came across an equation(HERE,pg 2 at the end) which was as follows: $$ xF''(x) +F'(x) + \frac{\omega^2}{g}F = 0 \hspace{1.5 cm} (A)$$ I was trying to convert it to the form $$x^2 y'' +xy' ...
1
vote
0answers
21 views

Reducing a system to first order

Convert the following to a first order system $$x''(t) = k_x(x(t) - y(t))^{-2}, \ \ y''(t) = k_y(x(t) - y(t))^{-2},$$ $$x'(0)=v_x, \ \ y'(0) = v_y, \ \ x(0) = x_0, \ \ y(0) = y_0.$$ I know how to ...
0
votes
1answer
29 views

Power series solution to $y' = y(1-y)$

Find the first five terms of the power series solution to the differential equation: $$y' = y(1-y)$$ Letting $y = a_0+a_1x+a_2x^2+a_3x^3+...$ It's evident that: $$y' = \frac{dy}{dx} = ...
3
votes
2answers
48 views

Properties of sin(x) and cos(x) from definition as solution to differential equation y''=-y

I recently came across the interesting definition of the sine function as the unique solution to the Initial Value Problem $$y'' = -y$$ $$y(0) = 0, y'(0) = 1$$ (My first question would be why this ...
0
votes
3answers
65 views

Solution of $xy\frac{dy}{dx} = \frac{(1+y^2)(1+x+x^2)}{1+x^2}$

Find the solution of $$xy\frac{dy}{dx} = \frac{(1+y^2)(1+x+x^2)}{1+x^2}$$ Please help me with this one. I have tried things like rewriting it as $$xy\frac{dy}{dx} = (1+y^2)+\frac{(1+y^2)x}{1+x^2}$$ ...
0
votes
1answer
19 views

Finding the solution to $u_{xx} - au_x = 0$

So as above, I am trying to find the general solution to: $$u_{xx} - au_{x} = 0$$ where $a \in \mathbb{R}$ Also, it should match the boundary conditions $$u(0,t) = 1, \quad u(1,t) = 0$$ It is not ...
0
votes
1answer
22 views

How to convert this into a bessel equation

I am trying to convert this into a Bessel equation of the form $$x^2y'' +xy' +(x^2 -n^2)y = 0$$ the equation is : $$ rR''(r) + R'(r) + \lambda^2r(R(r)) = 0 $$ Can you please indicate what type of ...
0
votes
2answers
27 views

Stability of equilibrium of first order ODE

Consider the autonomous ODE $$x' = x(e^{-x}-x^2).$$ Determine the stability at the trivial equilibrium $x_1 = 0$. Show that $f$ has exactly one equilibrium $x_2$ in $(0, 1)$. Determine ...
0
votes
1answer
30 views

Difference between constants, arbitrary constants and variables in differential equation

The general solution of the differential equation $y''+\omega^2y=0$ can be written as: $$y=\alpha\cos{(\omega(t-c))}+\beta\sin{(\omega(t-c))}$$ Is it correct to say that: $\omega$ and $c$ are ...
0
votes
1answer
25 views

Second Order Accurate Interpolation

On a grid I am having the values of a physical quantity say for example Temperature, at the E,W,N,S and P node all of them being calculated using a second order discretization scheme. I want a second ...
1
vote
1answer
32 views

Why do periodic boundary conditions imply no negative eigenvalues?

Im looking at the boundary value problem $$\Theta''+\lambda\Theta=0$$ with periodic boundary conditions $\Theta(\theta)=\Theta(\theta+2\pi)$ and $\Theta'(\theta)=\Theta'(\theta+2\pi)$. For the case ...
-1
votes
0answers
28 views

Can't parse Laplace Transform [closed]

L{delta [t- (pi/2)] sin t} ends up as U_(pi /2)_e^-(t-pi /2)sin 2(t -pi /2) and for the life of me I can't figure out how. Help?
0
votes
2answers
33 views

Solve for Harmonic Function $u_{xx} + u_{yy} = 0$

Solve the following problem: $u_{xx} + u_{yy} = 0$ for $(x,y)\in\{(x,y) | \sqrt{x^2+y^2}\lt1\}\cup\{(x,y)\in x,y\gt0\}$ $u(r,0) = u(0,r) = 0$ for $r \in (0,1)$, $u(\cos\theta,\sin\theta) = ...