All Questions
10
questions
0
votes
0
answers
47
views
Simplification of series (A good approximation will also work)
I have solved a system of linear ODEs and obtained a solution of the form
$
y_{ij}(t)=\sqrt{\frac{\gamma^{i(i-1)}}{\gamma^{j(j-1)}}}(\frac{-2\epsilon}{\alpha})^{i-j} \sum_{h=j}^i\frac{e^{-\frac{\alpha ...
- 61
1
vote
1
answer
87
views
Evaluate $\lim_{j \rightarrow +\infty} (I + A/j)^j$
Let $A$ be a $n\times n$ matrix. Evaluate
$$
\lim_{j \rightarrow +\infty} \left(I + \frac{A}{j}\right)^j.
$$
My guess is $e^A$.
My attept:
\begin{align*}
\lim_{j \rightarrow +\infty} (I + \frac{...
- 2,316
0
votes
1
answer
51
views
polynomial solution of second order differential equation
Find the polynomial solution
$$u_n(x) = x^n + a_1x^{n-1}+...+a_n$$
of the differential equation $$u_n'' + xu_n' - nu_n = 0$$ satisfied by u_n(x).
Note that this is entry-level calculus, so in my ...
- 1,334
2
votes
3
answers
505
views
Find a particular solution of $\,\,y''+3y'+2y=\exp(\mathrm{e}^x)$
I already solved for the homogeneous one, but I'm still looking for the particular solution of the differential equation:
$$y''+3y'+2y=\exp(\mathrm{e}^x)$$
The homogeneous solutions of this system ...
- 984
1
vote
3
answers
733
views
Find solution of differential equation $y'(t)=-2y(t)+1$
Could you help me explain how to find the solution of the differential equation
$$
y'(t)=-2y(t)+1,
$$
with
$$y(0)=1.$$
I know that the solution is
$$y(t)=\frac12 (1+e^{-2t}).$$
How about the IVP
$$...
- 519
0
votes
1
answer
499
views
An ODE with boundary conditions at infinity
I have a problem where:
$\ddddot{x} - 2 \ddot{x} + x = 0$
With boundary conditions
$x(0) = 1, \dot{x}(0) = 2, x(\infty) = 0, \dot{x}(\infty) = 0$
So I get my characteristic equation:
$s^4 -2s^2 + ...
- 227
0
votes
1
answer
28k
views
Find the general solution of $\,y'' + 9y = 0$
$y'' + 9y = 0\,$ and $\,y(0) = 0, \; y'(0) = 3.$
Since this has real roots, I use the general solution
$y_c = C_1 \mathrm{e}^{r_1 t} + C_2 \mathrm{e}^{r_2 t}$
I find the $y_c = \frac{1}{2}\mathrm{e}...
4
votes
3
answers
1k
views
Prove that if $\phi'(x) = \phi(x)$ and $\phi(0)=0$, then $\phi(x)\equiv 0$. Use this to prove the identity $e^{a+b} = e^a e^b$.
I am given the following. hint Consider $f(x)=e^{-x} \phi(x)$.
I am unsure how to approach this problem.
- 2,649
6
votes
1
answer
411
views
Solution of the IVP: $\,y'=\mathrm{e}^{-y^2}-1,\, y(0)=0$
Consider the initial value problem
$$
\frac{dy}{dx} = \mathrm{e}^{-y^2} - 1,\quad y(0)=0.
$$
The Method of Separation of Variables provides that:
$$
\int \frac{dy}{e^{-y^2} - 1} = x+c.
$$
I would ...
- 2,375
0
votes
1
answer
414
views
Commuting Exponential Matrices
Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$.
Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal.
$A, B$ square matrices.
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