To state: The solution of the given initial value problem using the method of Laplace transforms. Given: The initial value problem is, {y}text{}+{6}{y}'+{5}{y}={12}{e}^{t},{y}{left({0}right)}=-{1},{y}'{left({0}right)}={7}

Need help understanding the scaling properties of the Legendre transformation
At the wikipedia page for the Legendre transformation, there is a section on scaling properties where it says
${\displaystyle f\left(x\right)=ag\left(x\right)\to f\cdot \left(p\right)=ag\cdot \left(\frac{p}{a}\right)}$
and ${\displaystyle f\left(x\right)=g\left(ax\right)\to f\cdot \left(p\right)=g\cdot \left(\frac{p}{a}\right)}$
where f*(p) and g*(p) are the Legendre transformations of f(x) and g(x), respectively, and a is a scale factor.
Also, it says:
"It follows that if a function is homogeneous of degree r then its image under the Legendre transformation is a homogeneous function of degree s, where ${\displaystyle \frac{1}{r}+\frac{1}{s}=1}$.
1) I don't see how the scaling properties hold. I'd appreciate if someone could spell this out for slow me.
2) I don't see how the relation between the degrees of homogeneity (r,s) follows from the scaling properties. Need some spelling out here too.
3) If the relation ${\displaystyle \frac{1}{r}+\frac{1}{s}=1}$ is true, then could this be used to prove that linearly homogeneous functions are not convex/concave? (Because convex/concave functions have a Legendre transformation, and ${\displaystyle r=1}$ would imply ${\displaystyle s=\mathrm{\infty }}$, which is absqrt and thus tantamount to saying that a function with ${\displaystyle r=1}$ has no Legendre transformation. No Legendre transformation then implies no convexity/concavity.)

To sketch:
(i) The properties,
(ii) Linear transformation.
Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be the linear transformation that reflects each point through the
$x_{1}axis.$
Let $A=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

For large value of n, and moderate values of the probabiliy of success p (roughly, $0.05\Leftarrow p\Leftarrow 0.95$), the binomial distribution can be approximated as a normal distribution with expectation mu = np and standard deviation
$\sigma =\sqrt{np(1-p)}$. Explain this approximation making use the Central Limit Theorem.

The graph ${\displaystyle y=-2(\frac{3}{2}-e^{3-x})}$ by:
a) Performing the necessary algebra so that the function is in the proper form (i.e., the transformations are in the proper order).
b) Listing the transformations in the order that they are to be applied.
c) Marking the key point and horizontal asymptote.

Define the term Linear Transformation.

Find and describe an example of a matrix with each of the following properties.
Briefly describe why the desired property is in the matrix you picked. If no such matrix exists, explain using a theorem studied.
(a) A square matrix representing an injective but not a surjective transformation.

Linear transformation and its