New answers tagged

0 votes
Accepted

what more can I say using Sturm's comparison theorem?

Sturm's comparison theorem is not the correct approach. If we set up $x(t) = \sqrt{x}u(\frac{x^2}{2})$, then $u$ satisfies Bessel equation. Using the asymptotic decay of Bessel function we can ...
alejandro's user avatar
  • 113
0 votes
Accepted

differential equation $1+x\sqrt{(x^2-y^2) } - (y\sqrt{(x^2-y^2)})y'=0$

$$1+x\sqrt{(x^2-y^2) } - (y\sqrt{(x^2-y^2)})\frac{dy}{dx}=0$$ $$\implies 1+\sqrt{x^2-y^2}\bigg(x-y\frac{dy}{dx}\bigg)=0$$ Let $t={x^2-y^2},\frac{dt}{dx}=2(x-y\frac{dy}{dx})$. So$$1+\sqrt t\bigg(\frac{...
Gwen's user avatar
  • 1,427
0 votes

Jacobian matrix from mass action kinetic ODE network

To follow up on my comment Denote the unusual power function by $y=x^A$ and calculate the differentials of the elementwise functions $$\eqalign{ \def\qiq{\quad\implies\quad} w &= \log(x) &\qiq ...
greg's user avatar
  • 36.1k
2 votes
Accepted

How does one do this translation?

I have found the original paper you've taken the screenshot from and here's the deal. Your approach is absolutely correct, you shouldn't be confused with the constant terms because later on they are ...
Egor Larionov's user avatar
-1 votes

Diferencial Equations solve $\frac{e^{2x}+2x}{e^{x}}dy-(1+2y)dx=0$

This function can't be integrated .Only a part of this equation can be integrated and not the full .Note that many functions like this can't be integrated or anti derivative can't be found .This can ...
Praneet.S 's user avatar
0 votes

Asymptotic Expansion of Integrals via Integration by Parts - Leading Behavior

$$\int_1^\infty \frac{\cos(xt)}{t}\,dt=-\text{Ci}(x)$$ Expanded as series $$\int_1^\infty \frac{\cos(xt)}{t}\,dt=-\log (x)-\gamma+\sum _{n=1}^\infty (-1)^{n+1} \frac {x^{2n}}{ 2n\,(2n)! }$$
Claude Leibovici's user avatar
1 vote
Accepted

Diferencial Equations solve $\frac{e^{2x}+2x}{e^{x}}dy-(1+2y)dx=0$

I am afraid that only a series solution would give a result. For example (using @Travis Willse's suggestion in comments), this would give $$y=C +(2C+1)\sum_{n=1}^\infty (-1)^{n+1}\, a_n\, x^n$$ the ...
Claude Leibovici's user avatar
0 votes

Jacobian matrix from mass action kinetic ODE network

Answer: Thank you to @LutzLehmann for the suggestion, which pointed me to the solution. The Jacobian can be split into two parts: $$ J = S T'(x) $$ with $S = (B-A)^T$ (the stochiometry matrix) and $T'(...
hr8tpa's user avatar
  • 11
2 votes

Periodic perturbation of ODE

For simplicity, set the equilibrium to be $0$ and $A=DF(0)$. So $$ F(x) = Ax+O(x^2). $$ Thus the DE becomes $$ x'=Ax+\epsilon f(t)+ O(x^2).$$ Choose the constant $\delta$ to satisfy $$ \int_0^\theta e^...
xpaul's user avatar
  • 44.1k
2 votes
Accepted

Prove that two integrating factors define a solution.

Differentiating $\mu(x,y)=cv(x,y)$ with respect to $x$, one obtains $$ \mu_x+\mu_yy'=c(v_x+v_yy')=\frac{\mu}{v}(v_x+v_yy') $$ $$ \implies y'=\frac{v\mu_x-\mu v_x}{\mu v_y-v\mu_y}. \tag{1} $$ On the ...
Gonçalo's user avatar
  • 9,447
1 vote

Help on transformation of boundary conditions

I'm not sure, but my guess is that in Eq. (3.14), $y$ is interpreted as a function of $x$, so that $y'$ is the derivative of $y$ with respect to $x$ (and not $\eta$): $y'(x) = \partial_x y(x)$. With ...
user1302959's user avatar
3 votes
Accepted

How to solve $2xy\frac{dy}{dx}+(1+x)y^2=e^x$?

The equation can be multiplied by $\mathrm{e}^{x}$ to become $$2x\mathrm{e}^{x}y\frac{\mathrm{d}y}{\mathrm{d}x}+(1+x)\mathrm{e}^{x}y^2=\frac{\mathrm{d}}{\mathrm{d}x}\left(x\mathrm{e}^xy^2\right)=\...
Sai Mehta's user avatar
  • 1,161
1 vote

Weak form FEM discretisation of Non-Linear System

Let $\langle\cdot,\cdot\rangle$ denote the inner product on $L^2(0,1)$, then we do the following steps Form the residual $F(u) = 0$. Approximation $u$ and substitute: $F\left(\sum_{i=0}^N u_i \phi_i\...
whpowell96's user avatar
  • 5,632
0 votes

Is there a formula for the Butcher tableau for the implicit Runge Kutta method for an arbitrary number of stages?

Yes ! Well, more or less. You'll find this in the following book for collocation methods (like Gauss). Hairer, Ernst; Lubich, Christian; Wanner, Gerhard, Geometric numerical integration. Structure-...
G. Fougeron's user avatar
  • 1,544
2 votes

Why is the uniqueness theorem for 1st order differential equations involve taking a partial dervative?

As peE_ pointed out Picard-Lindelöf theorem requires Lipschitz continuity for the unique existence of the solution. If the function is not Lipschitz continuous then such existence is not guaranteed. ...
ioveri's user avatar
  • 1,492
2 votes

Why is the uniqueness theorem for 1st order differential equations involve taking a partial dervative?

Let $f:\Omega \subset \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$, $x'(t) = f(t,x)$ with initial condition $(t_{0},x_{0})$. For Picard-Lindelöf theorem it is needed $f(t,x)$ to be ...
peE_'s user avatar
  • 21
0 votes

How to solve simple second order ODE with RHS $x / \sigma^2$

By inspection, it may be seen that both $$ x_i(t)=e^{t/\sigma_i},\quad x_i(t)=e^{-t/\sigma_i} $$ are solutions to the equation. Hence the general solution is a linear combination of the form $$ x_i(t)=...
CW279's user avatar
  • 799
0 votes

Transition probability density function for a non-trivial diffusion process.

We provide an additional solution, i.e. the eigenfunctions of the infinitesimal generator, when $\beta_1 - 1 = \theta (\beta_2-1)$ and $\theta=2 {\tilde \theta}$ where $-{\tilde \theta} \in {\mathbb N}...
Przemo's user avatar
  • 11.4k
1 vote

Help with solving an ODE involving nested square roots

Continuing with your substitution: $$\sqrt{\alpha+\left|\cos(\phi)\frac{d\phi}{dx}\right|}=\frac{1}{\beta}\cos(\phi)$$ Rearranging: $$\frac{\cos(\phi)}{\cos^2(\phi)-\alpha\beta^2}d\phi = \pm\frac{1}{\...
Etotheipi's user avatar
  • 128
1 vote
Accepted

How do I check if set of solutions to differential equation form vector subspace

You can apply the usual characterisation of subspaces: Check that $0$ is a solution to the equation; then take any solutions $x(t), y(t)$ (no need to find them explicitly, work formally) and verify ...
Julio Puerta's user avatar
  • 4,931
0 votes

How do I check if set of solutions to differential equation form vector subspace

Like with any subspace question you have to make sure , a)zero is in it, with f, r*f is in it and with f and g f+g are in it.
trula's user avatar
  • 1,522
0 votes

Finding the coefficients of a general solution to an homogeneous ODE via Sturm-Liouville theory

HINT: Let $y(x)=Y(\ln x)$, which is equivalent to $Y(x)=y(e^x)$. Then $y'(x)=Y'(\ln x)\frac{1}{x}$, so that $xy'(x)=Y'(\ln x)$. This gives an equivalent equation $$ x\frac{d}{dx}\left(x\frac{dy}{...
Disintegrating By Parts's user avatar
1 vote

Best way to decouple a system of second-order ODE

Adding both equations as real and imaginary part gives $$ z''=x''+iy''=\Omega(y'-ix')+\omega_0^2(x+iy) \\=-i\Omega z'+\omega_0^2z $$ The term coupling the real and imaginary parts is the first-order ...
Lutz Lehmann's user avatar
4 votes
Accepted

Writing a general solution to differential equation with Bessel functions

Applying the substitution $s = \mathrm{e}^{-\alpha t / 2}$ in the hint transforms the differential equation to $$s^2 x_{ss} + \left(1-\frac{2 \beta}{\alpha}\right) s x_s + \frac{4 \omega_0^2}{\alpha^2}...
Travis Willse's user avatar
2 votes

Solution to Brachistocrone with friction

Differentiating the equation $$ y-\mu x = \frac{1}{C}\left(\sin\frac{\theta}{2}+\mu\cos\frac{\theta}{2}\right)^2 \tag{1} $$ with respect to $\theta$, one obtains $$ \dot{y}-\mu\dot{x}=\frac{1}{C}\left(...
Gonçalo's user avatar
  • 9,447
1 vote
Accepted

Give an example of an origin-passing function that is continuous not Lipschitz and its integral on a interval $[0,b]$, $b>0$ is infinite

Take $f(x)=-x\log x$, it is not Lipschitz at 0 but has infinite integral.
Liding Yao's user avatar
  • 1,815
0 votes

Question about lower bound when solving Variation of Parameter IVPs

The general solution to $y'-2xy=2$ is $$ y=e^{x^2}\left(C+2\int_a^xe^{-s^2}ds\right), \tag{1} $$ where both $C$ and $a$ are arbitrary constants. Since the initial value of $y$ is defined at $x=0$, it ...
Gonçalo's user avatar
  • 9,447
1 vote

Help Solving Second-Order ODE

Maple gives a solution in terms of modified Bessel functions: $$y \! \left(x \right) = {\mathrm e}^{\frac{-b^{2} x +a \,{\mathrm e}^{-b x}}{2 b}} \left(-K_{\frac{-b +2 \,\mathrm{i}}{2 b}}\! \left(-\...
Robert Israel's user avatar
0 votes
Accepted

Recovering a linear ODE w/ constant coefficients given the roots of the auxiliary equation

Hint: Hint 2: Sketch of solution:
Sean Roberson's user avatar
0 votes

Solving Systems of Non-Homogenous ODE that Have Problems with Linear Independence

The solution to the differential equation of the form $\dot{x}=Ax+bv$ with initial condition $x(t_0)=x_0 $is given by $$ x=e^{A(t-t_0)}x_0+\int_{t_0}^t e^{A(t-\tau)}bvd\tau $$ for $x \in \mathbb{R}^{n}...
Eddy Y's user avatar
  • 441
3 votes
Accepted

Backwards Euler on System of Equations

The problem appears here: x2(n+1) = x2(n) + h*(-5x2(n+1)-3x1(n+1) + sin(pi*t(n+1)/4) (1-5h)x2(n+1) = x2(n) + h*(-3x1(n+1) + sin(pi*t(n+1)/4) When you move ...
Egor Larionov's user avatar
8 votes

Does ODE solution implies existence for a proportional ODE?

It's not true. Let $$ f(x,t) = \cases{1 & if $x \le t + 1$\cr 1 + (x-t-1)^2 & otherwise\cr}$$ The solution of $x' = f(x,t)$ with initial condition $x(0)=0$ is $x=t$, defined for all real $t$. ...
Robert Israel's user avatar
0 votes
Accepted

How to solve the diferential equation $(1+t^2)y' + ty^3=0$

It's a separable ode, as stated by @Chinny84 . HINT $$ y^{-3} dy = \frac{-t}{1+t^2} dt $$ Now integrate on both sides. Assume $u=1+t^2 \implies du=2t \ dt $ $$\int y^{-3} \ dy = \int \frac{-du}{2u} \ ...
Gwen's user avatar
  • 1,427
6 votes
Accepted

How to calculate $\int(yy'' + (y')^2)\,dx$?

Even if you couldn't have noticed the $vu' + uv'$, you could still do through integration Integrating through by-parts: \begin{align*} \int yy'' + (y')^2 \, dx &= \int yy'' \, dx + \int (y')^2 \, ...
OpateItZOpatoOpate's user avatar
1 vote
Accepted

$f f'' \geq 2(f')^ 2 \implies f$ decreasing

$f$ is strictly convex hence $f'(x)$ is strictly increasing. We need to show that it is always negative. If for some $c\ge 0$ and $x_0$ we have $f'(x_0)\ge c$ then $f'(x)>c$ for all $x>x_0$. ...
van der Wolf's user avatar
  • 2,370
0 votes

Verifying the Solution to a 2nd Order ODE/Finding $y''$

Just adding onto Dr Graubner's answer above, it can help if you simplify the second derivative a bit further: $$ y''=\tan{x}+\cos{x} \ln\left(\frac{1+\sin{x}}{\cos{x}}\right) $$ Apologies if this not ...
jonathandavis's user avatar
0 votes

Finding mean first passage time for reflecting Brownian motion

Define $\tau_x=\inf \{t\ge 0: X_t\in \partial B_{\epsilon}(0)|X_0=x\}$ and $G(x,t)=P(\tau_x>t)$. Using the backward equation involving the transition kernel for a time-homogenous process, i.e., $p(...
Fellow InstituteOfMathophile's user avatar
1 vote

Transition probability density function for a non-trivial diffusion process.

Here we will only focus on finding the eigenfunctions of the infinitesimal generator. In other words we want to solve the following ODE below: \begin{equation} \left(\mu z^{\beta_1} \frac{d}{d z} + \...
Przemo's user avatar
  • 11.4k
2 votes

On the solutions of $y''=yP(x)$ or $y'+y^2=P(x)$

I looked up the four coefficient sequences of the sums in the OEIS. The first sum is elementary and the other three can be expressed as ${}_2F_1$'s: $$\sum_{n=0}^\infty\binom{3n}nx^n=\sum_{n=0}^\infty\...
Parcly Taxel's user avatar
0 votes

Solve ODE: $y'(x)=y(x)^2+x^2$

Let $y=-z'/z$, then $z$ satisfies $z''+x^2z=0$ whose linearly independent solutions in the notation of DLMF 12.2.3 are $W(0,\pm\sqrt2x)$. By DLMF 12.14.13 another linearly independent pair of ...
Parcly Taxel's user avatar
1 vote
Accepted

How to solve a differential equation of the 3rd order without factorisation

$$\dddot{y} + m\ddot{y} - 2m^3y = 0$$ This is ambiguous because the argument of the function $y(?)$ is not defined. Is it $y(x)$ or $y(m)$ or $y(t)$ or what else? The differentials are wrt what ...
JJacquelin's user avatar
  • 66.3k
2 votes

Approximating solution of a second degree ODE

I think the following is a proof for that $\lim_{x \to \infty} y'(x)$ exists. Maybe you can continue with this. W.l.o.g let $p$ and a solution $y$ be defined on $[1,\infty)$. Integrating twice yields $...
Gerd's user avatar
  • 7,099
0 votes
Accepted

Special homogeneous linear ODE of the second order

Hint. One solution to the ODE can be found by inspection: it's $y_1(x)=Cx$. Another solution can be found with the ansatz $y_2(x)=xv(x)$. Indeed, substituting the latter in the ODE, we obtain $$ xv''+...
Gonçalo's user avatar
  • 9,447
0 votes

Dimension of solution space

Really good question. I used to have this doubt before. The space of the solutions of $y''+y = 0$ is: $$ S:=\{a_1 cos(x) + a_2 sin(x)|a_1,a_2 \in \mathbb{R}^1\} = span(cos(x),sin(x)) $$ So the ...
Bruce Lee of USTC's user avatar
1 vote

Proving $\int\limits_{Q} u^2 dx ≤ C(n) \int\limits_{Q} (u^2_{x1} + · · · + u^2_{xn}) dx$ over a cube $Q$

Let $V(x)=x=(x_1, \dots, x_n)$, then we get by the divergence theorem and the fact that $u$ vanishes on $\partial Q$ $$\int_Q u(x)^2dx = n^{-1} \int_Q u(x)^2 \text{div}(V)(x)dx =- \int_Q \nabla(u(x)^2)...
Severin Schraven's user avatar
0 votes

Proving $\int\limits_{Q} u^2 dx ≤ C(n) \int\limits_{Q} (u^2_{x1} + · · · + u^2_{xn}) dx$ over a cube $Q$

One possible proof would be via developing $u$ into its Fourier series and then using simple formula for second norm (so called Parselval's theorem). In case of $n=1$, say $u = \sum a_ke^{2{\pi}kx}$, ...
Salcio's user avatar
  • 117
0 votes

An differential equation question that seems exact but not

We rearrange the terms to render $[(x)dy+(y)dx]+[x dx]-[y dy] = 0.$ Then the first term in brackets is just tge differential of $xy$ by the Product Rule, and the other terms are one-variable ...
Oscar Lanzi's user avatar
  • 39.6k
0 votes

An differential equation question that seems exact but not

You are correct to think that is an exact DE, the only obstacle on your way is that the total differential has undergone arithmetic transformations. To see this, let us solve the second equation and ...
Egor Larionov's user avatar
1 vote

Solve $\frac{d}{dx}f(x)=f(x-1)$

Using the Laplace transform $$ s \hat f(s) = e^{-s}\hat f(s) + e^{-s}\int_{-1}^0e^{-sx}f(x)dx+f_0 $$ Here $f(x),x\in[-1,0)$ is considered known, kind of initial conditions. Then the Laplace transform ...
Cesareo's user avatar
  • 33.4k
0 votes
Accepted

Solving a differential equation with initial conditions using Laplace transforms

Note that $$\mathcal L \{te^{-3t}\}=\left.\mathcal{L}\{t\}\right|_{s\to s+3}=\frac{1}{(s+3)^2}$$ So instead we get: $$ \begin{align} s^2Y(s)+7sY(s)+10Y(s) = \frac{4}{(s+3)^2}-1\implies y(x) = \mathcal ...
Masd's user avatar
  • 643

Top 50 recent answers are included