Intereting Posts

Maximum number of edges in a non-Hamiltonian graph
Are the matrix products $AB$ and $BA$ similar?
Show that $g$ is injective
Show adding rows to a non-singular square matrix will keep or increase its minimum singular value
Integrate product of Dirac delta and discontinuous function?
Existence of least squares solution to $Ax=b$
Is every flat manifold with boundary locally isometric to the Euclidean half-space?
How find this $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\zeta_{n}(3)}{n}=?$
Why are affine varieties except points not compact in the standard topology on $C^n$ ?
Inverse Laplace of $ \frac{1}{\sqrt{s} – 1} $?
The Gaussian and Mean Curvatures of a Parallel Surface
Expected number of steps to transform a permutation to the identity
Inverse image sheaf and éspace étalé
Evaluation of $\int_{0}^{1} \frac{dx}{1+\sqrt{x}}$ for $n\in\mathbb{N}$
Finding the basis of $\mathfrak{so}(2,2)$ (Lie-Algebra of $SO(2,2)$)

Convert the ODE system

$$

\dot{x}=\begin{pmatrix}a(t) & b(t)\\c(t) & d(t)\end{pmatrix}x

$$

into polar form. You should get two equations

$$

\frac{d}{dt}\Phi(t)=…\\ \frac{d}{dt}\ln r(t)=….

$$

I set

$$

x_1:=r(t)\cos\Phi(t)\\ x_2:=r(t)\sin\Phi(t)

$$

and got

$$

\frac{d}{dt}\Phi(t)=b(t)+\frac{\frac{d}{dt}r(t)\cos\Phi(t)}{r(t)\sin\Phi(t)}-\frac{a(t)\cos\Phi(t)}{\sin\Phi(t)}\\ \frac{d}{dt}\ln r(t)=d(t)+\frac{c(t)\cos\Phi(t)}{\sin\Phi(t)}-\frac{\cos\Phi(t)\frac{d}{dt}\Phi(t)}{\sin\Phi(t)}

$$

Would like to know if this is right.

- Bessel Equations Addition Formula
- Solving second-order ODE using an integrating factor
- Linearization of PDE: $0$ is an eigenvalue since all translates of travelling waves are also travelling waves
- Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?
- system of differential linear equations $y'=\begin{pmatrix}1 & 1\\0 & 1\end{pmatrix}y$
- Manifold interpretation of Navier-Stokes equations

With greetings

- Is this a correct/good way to think interpret differentials for the beginning calculus student?
- Solve this system of equations using elimination for $x(t)$ and $y(t)$
- Numer of solutions for IVP
- ODE when the nonhomogeneity is complex
- Solve $(x^2 + 1)y'' - 6xy' + 10y =0$ using series method
- Not empty omega limit set
- Using this trick to solve an integro-differential equation
- Runge Kutta with Impulse
- Fourth Order Nonlinear ODE
- Solving a challenging differential equation

I have the following:

$$x_1:=r(t) \cos \Phi(t),$$

$$x_2:=r(t) \sin \Phi(t)$$

and the derivatives are:

$$\dot{x_1}=\dot r \cos \Phi – r \sin(\Phi)\, \dot \Phi,$$

$$\dot{x_2}=\dot r \sin \Phi + r \cos(\Phi)\, \dot \Phi.$$

with $\dot x \equiv \dfrac{d x}{dt}.$

So you get:

$$\dot r \cos \Phi – r \sin(\Phi)\, \dot \Phi =a r\cos \Phi+br\sin\Phi,$$

$$\dot r \sin \Phi + r \cos(\Phi)\, \dot \Phi =c r\cos \Phi+r d\sin\Phi.$$

The equation for $\dot\Phi$ is:

$$\dot\Phi =\dfrac{(a r\cos \Phi+br\sin\Phi)\sin\Phi-(c r\cos \Phi+rd\sin\Phi)\cos\Phi}{-r},$$

and for $\dot r$ is :

$$\dot r = (a r\cos \Phi+br\sin\Phi)\cos\Phi – (c r\cos \Phi+rd\sin\Phi)\sin\Phi, $$

but:

$$\dfrac{\dot r}{r(t)} = \dfrac{d}{dt}\ln[r(t)],$$

so:

$$\dfrac{d}{dt}\ln[r(t)]=(a \cos \Phi+b\sin\Phi)\cos\Phi – (c\cos \Phi+d\sin\Phi)\sin\Phi. $$

- Moment of inertia about center of mass of a curve that is the arc of a circle.
- What is the meaning of $dz$ in Complex Integrals?
- Visual explanation of $\pi$ series definition
- Geometry formulas, how to show identities.
- Prove that $\sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$
- Lower bound for $\phi(n)$: Is $n/5 < \phi (n) < n$ for all $n > 1$?
- Number of permutations of thet set $\{1,2,…,n\}$ in which $k$ is never followed immediately by $k+1$
- On the Definition of multiplication in an abelian group
- Prove that $\frac{1}{2}+\frac{2}{2^2}+\frac{3}{2^3} +…+ \frac{n}{2^n} = 2 – \frac{n+2}{2^n} $
- How to find logarithms of negative numbers?
- Generated $\sigma$-algebras with cylinder set doesn't contain the space of continuous functions
- When Are We Allowed to Break Up A Triple Integral?
- Will $\sum^{\infty}_{n=1}|a_n|^p$ converge?
- Differentiation with respect to a matrix (residual sum of squares)?
- Which smooth 1-manifolds can be represented by a single smooth parametrization?