Intereting Posts

How do we show that $\int_{0}^{1}{\ln^k(x)\over x}\ln\left(1-\sqrt{x}\right)\mathrm dx=(-n)^{k+1}k!\zeta(k+2)?$
What are some motivating examples of exotic metrizable spaces
Prove $\lim_{j\rightarrow \infty}\sum_{k=1}^{\infty}\frac{a_k}{j+k}=0$
Equation that can easily be changed to output the digit in 1's, 10's ,100's etc?
Proving that if two linear transformations are one-to-one and onto, then their composition is also.
The Three Princesses (distinguishing truth-teller with 1 question)
How to evaluate the integral $\int e^{x^3}dx $
How to get closed form solutions to stopped martingale problems?
On critical points of a large function
Does someone know why raising the element of a group to the power of the order of the group yields the identity?
What does it mean for the group of rotation matrices to have a “manifold structure”
Find the value of $\lim_{x \to – \infty} \left( \sqrt{x^2 + 2x} – \sqrt{x^2 – 2x} \right)$
Why an eigenspace is a linear subspace, if the zero vector is not an eigenvector?
Robertson-Seymour fails for topological minor ordering? (I.e., subgraphs and subdivision)
Probability of 3 Heads in 10 Coin Flips

Determine wether the differential equation is stable or not for the following conditions:

$x”'(0) = x”(0) = x'(0) = x(0) =0$

$\frac{d^3x(t)}{dt^3}+\frac{d^2x(t)}{dt^2}+4\frac{dx(t)}{dt}+4x(t) = e^t(H(t)-H(t-3))$

- If an IVP does not enjoy uniqueness, then there are infinitely many solutions.
- Solutions of homogeneous linear differential equations are a special case of structure theorem for f.g. modules over a PID
- General Solution for a given system of equations
- What's so special about sine? (Concerning $y'' = -y$)
- Counter-example to Cauchy-Peano-Arzela theorem
- system of ode with non-constant coefficient matrix

Now, I got that the solution to this differential equation reduces x(t) by use of Laplace Transforms, however, how do I determine whether or not it is stable?

$x(t) = 0.1e^{-t} + 0.1e^t – 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})*H(t-3)$

Now I first thought I would look at it graphically and I used Matlab software to get a plot as shown below:

And from the plot above it clearly appears to diverge (though this is not a proof).

Anyone have an idea as to an elegant proof to show the solution is unstable?

Anyone help would be greatly appreciated!

- What's the difference between an initial value problem and a boundary value problem?
- Green's function in a moving frame for a constant heat source
- Why it is absolutely mistaken to cancel out differentials?
- Find the trajectories that follow drops of water on a given surface.
- Exact Differential Equations
- Solve Boundary Value Problem for $y''+ y' + e^xy = f(x)$
- What kind of book would show where the inspiration for the Laplace transform came from?
- Time required to reach the goal when an object will be slowing down incrementally based on distance travelled?
- Differential Equations: solve the system
- Generalized “Worm on the rubber band ” problem

You have eigenvalues $-1,2i,-2i$ and no forcing after $t=3$. This is stable, but not asymptotically stable, i.e., solutions do not shrink to zero, but they also do not grow.

As you found, the homogeneous solution, which is valid for $t>3$, is

$$

x(t)=c_1\cos(2t)+c_2\sin(2t)+c_3e^{-t}

$$

where the first two components stay bounded while the third converges towards zero.

That a formula represents a stable solution does not automatically mean that the evaluation of that formula is numerically stable. Indeed, for large $t$ the floating point evaluation of that formula reduces to the evaluation of $0.1·(e^t-e^3·e^{t-3})$ which can incur cancellation error of the size $0.1·e^t·10^{-16}$ which for $t=100$ gives the magnitude $10^{-1+43-16}=10^{26}$ which is indeed also the scale factor in the graph.

One needs to explicitly switch off this exponential factor via

$$

0.1·e^t·(H(t)-H(t-3))

$$

or a similar form of conditional evaluation.

- $f_n(x) = x – x^n$ for $x\in $. Does the sequence converge pointwise or uniformly on $$?
- The index of nilpotency of a nilpotent matrix
- Proving : $ \bigl(1+\frac{1}{n+1}\bigr)^{n+1} \gt (1+\frac{1}{n})^{n} $
- Deriving Fourier inversion formula from Fourier series
- Factor group of a center of a abelian group is cyclic.
- Calculating $\lim_{n\to\infty}\sqrt{n}\sin(\sin…(\sin(x)..)$
- Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?
- Roundest ellipse with specified tangents
- Is the dihedral group $D_n$ nilpotent? solvable?
- Proving $E=3σ^4$
- Stuck in integration problem
- Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-1$
- How to win at roulette?
- Strong Counterexample to MVT on Q
- Prove that the centre of the nine-point circle lies on the midpoint of the Euler line