Homework Statement
I have
J=\begin{bmatrix}
\frac{\pi}{2}&0&0\\
1&\frac{\pi}{2}&0\\
0&1&\frac{\pi}{2}\\
\end{bmatrix}
I need to find \sin(J) \text{ and } \cos(J) \text{ and show that } \sin^{2}(J)+\cos^{2}(J)=I
Homework Equations
The Attempt at a Solution
I have the...
ha ha , it required puzzling, because I knew that
p(x)=\sum_{i=1}^{n}p(\lambda_{i})h_{i}(x)
would have to be such that
p(A)=2^{A}
but because of a silly mistake of writing a 0 instead of a 1, I was not getting that conclusion and so thought all my work was incorrect.
After much puzzling, I believe I have the answer:
2^{A}=2E_{1}+2^{2}E_{2}
where I have found
E_{1}=\begin{bmatrix} 1&0&0\\1&1& \-1\\1&0&0\end{bmatrix}
E_{2}=\begin{bmatrix}0&0&0\\-1&0&1\\-1&0&1\end{bmatrix}
so I have
2^{A}=\begin{bmatrix}...
Homework Statement
Find the spectral decompostion of
$$
A=
\begin{matrix}
1 & 0 & 0\\
-1 & 1 & 1 \\
-1 & 0 & 2
\end{matrix}
$$
and use this to find $$ 2^{A} $$
Homework Equations
The Attempt at a Solution
I have found the eigenvalues to be : $$ \lambda_{1}=1 \text{...
\int(\frac{1}{ur}+\frac{1}{u(u-r)})
=\frac{1}{u}\int\frac{1}{r}+\int\frac{1}{u-r}
=\frac{1}{u}(ln(r)-ln(u-r))
=\frac{1}{u}(ln\frac{r}{u-r})
my problem is that i don't know where to put r_0 here. The solution has r_0
Homework Statement
i need to integrate:
\frac{1}{r(u-r)}dr
Homework Equations
u is a constant
The Attempt at a Solution
im not sure if i should decompose the fraction. i tried that, but it didnt seem to be of any help.
ok so i get
\frac{ln(r)-ln(r-u)}{u}=t
which i then manipulate (or try to at least ) to get:
ln\frac{r}{r-u}=ut
if this is correct then i think i can do:
\frac{r}{r-u}=e^{ut}
i know something is wrong cos i don't have r_0
im working through an example in jordan and smith...
Homework Statement
\frac{dr}{dt}=r(u-r)
Homework Equations
i rewrote this as \frac{dr}{dt}-ur=-r^2
i think this is like bernoulli's equation
The Attempt at a Solution
so i let w=\frac{1}{r} so that r=\frac{1}{w}
this gives me
\frac{dr}{dt}=-1w^{-2}\frac{dw}{dt}
so now...
Homework Statement
u\frac{\delta u}{\delta x}+\frac{\delta u}{\delta y} =1
u|_{x=y}=\frac{x}{2}
Homework Equations
the characteristic equations to solve are:
\frac{dx}{ds}=u
\frac{dy}{ds}=1
\frac{du}{ds}=1
The Attempt at a Solution
I got the following equations from the...
hey,
so I am only just starting to understand method of characteristics, but I am gona try to help.
The first step I believe is to get your characteristic equations
\frac{dx}{ds}=x^{2}
\frac{dt}{ds}=1
\frac{du}{ds}=t
solving these equations gives
x=x^{2}s +x_{0}
t=s+t_{0}...