Differential Equations and Linear Algebra Q&A

What is the product of $3$ numbers?

Evaluate the expression $C25$?

Let $w=\frac{yz}{x}$ where $x=t^2,y=r+t$ and $z=r-<!--\; -\; -->t$. Find $\frac{\mathrm{\partial <!--\; \partial \; -->}w}{\mathrm{\partial <!--\; \partial \; -->}t}$ and $\frac{\mathrm{\partial <!--\; \partial \; -->}w}{\mathrm{\partial <!--\; \partial \; -->}r}$ by using Chain Rule.

Differentiate
a) $\frac{5x^2}{x^3+1}$
b) $\frac{1}{1+e^x}$
c) $\frac{e^x}{x^2}$

How would you differentiate
$\sec <!--\; \; -->x{e}^x$

Notation of differential equations
I have just started a course on differential equations, and unfortunetely enough for me, we immediately used notation foreign for me, for example:
$$x^2{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}^2=\sin <!--\; \; -->(x)\text{ and}y\times <!--\; \times \; -->{\displaystyle \frac{d^{2}y}{d{x}^2}}=\sin <!--\; \; -->(x)$$
were used as examples of non-linear ordinary differential equations. My questions are
- Is $\frac{dy}{dx}$ equal to y′? Also, what then is $\frac{d^{2}y}{d{x}^2}$?
I have looked up the formal mathematical definition but it somewhat confuses me,
- Why on earth is this notation used, instead of just using ′ or ′′? It seems to me much more confusing and unnecessarily messy.

Differential equations, money
The increase of money in your account is proportional to the amount of money that you already have in your account. You deposit $1 per hour into your account. Model this using differential equations.
My attempt:
$y^{\prime }=ky+a$ where a is the portion of that $1/h. But the question is, how am I supposed to find out the correct portion of that $1/h. That is,how do I find a?

Functional vs Differential Equations?
Background
I'm under the impression that a differential equation has locality hardwired into it (which is why we see them more in physics). However, if I wanted to write something non-local I'd use a functional equation. I am aware there are cases where the differential equation has a functional equation form as well.
Question
- Is my initial assumption true: "differential equation has locality hardwired into it"?
- Is the cardinality of the set of functional equations more than that of the differential equations?
- Is there any functional equation which does not have any differential equation equivalent?
- Is there any differential equation which does not have any functional equation equivalent?
Caveat
While different theories of physics have different definitions of locality (for example bell locality in quantum mechanics and micro-causality in QFT) they all agree one cannot send information (non-random message) from point A to point B without a mediator (which is what I mean by locality in the above in the above).

Find the general solution of the differential equation. $$\frac{dy}{dx}=10y$$

System of differential equations Method integrable combinations
$$dx/dt=x^2-<!--\; -\; -->2xy$$
$$dy/dt=2xy-<!--\; -\; -->y^2$$

Solve this differential equation
$$(2x+y)dx+(x-<!--\; -\; -->2y)dy=0$$
as an exact differential equation and I know it's exact because I solve the equaliy
$$\frac{\mathrm{\partial <!--\; \partial \; -->}(2x+y)}{\mathrm{\partial <!--\; \partial \; -->}y}=1$$
and
$$\frac{\mathrm{\partial <!--\; \partial \; -->}(x-<!--\; -\; -->2y)}{\mathrm{\partial <!--\; \partial \; -->}x}=1$$
so following the steps to solve this kind of equations i have:
$$x^2+g^{\prime }(y)=x-<!--\; -\; -->2y$$
and
$$g^{\prime }(y)=\frac{x-<!--\; -\; -->2y}{x^2}$$
to be honest I have many doubts what are the next steps so if you can guide me I'll apreciate

Exact Differential Equations
I was revising differential equations and came across the topic of exact differential equations. I have a doubt concerning it. Suppose the differential equation $M(x,y)dx+N(x,y)dy=0$ is exact. Then the solution is given by: $\int <!--\; \int \; -->Mdx+\int <!--\; \int \; -->(N-<!--\; -\; -->\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}y}\int <!--\; \int \; -->Mdx)dy=c$. I understand that the integrand in the second term is a function of y alone and also understand the derivation of this solution. What I don't understand is the following paragraph:
My book then says "Since all the terms of the solution that contain x must appear in $\int <!--\; \int \; -->Mdx$, its derivative w.r.t. y must have all the terms of N that contain x. Hence the general rule to be followed is: Integrate $\int <!--\; \int \; -->Mdx$ as if y were constant. Also integrate the terms of N that do not contain x w.r.t. y. Equate the sum of these integrals to a constant."

Solving Differential Equations w/ polynomials
I'm new to solving differential equations.
How would we go about solving a differential equation like
$y^{\prime }=-<!--\; -\; -->4+5y-<!--\; -\; -->y^2$?

differential equations and physical intuition
Often when you study differential equations, you find phenomena in nature modeled by those equations. Sometimes an insight into a physical problem can help you to solve a differential equation. My question is: If you are a pure mathematician studying differential equations, do you have to be good at physics (biology, finance) too?

Differential equations system
I have found the following example in one of my courses but I don't have any similar exercises resolved so I would like to know how to solve this:
The differential equations system is the following:
$$x_1^{\prime }=3x_1-<!--\; -\; -->2x_2+e^t$$
$$x_2^{\prime }=2x_1-<!--\; -\; -->x_2+2e^{2t}$$
a) Write the system in the matriceal form $x^{\prime }=Ax+b(t)$
b) Determinate the solutin of the system.

Division of differential equations
$$\frac{dx(t)}{dy(t)}=\frac{\mathrm{\alpha <!--\; \alpha \; -->}x(t)-<!--\; -\; -->\mathrm{\beta <!--\; \beta \; -->}x(t)y(t)}{-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}y(t)+\mathrm{\delta <!--\; \delta \; -->}x(t)y(t)}$$
How would one simplify this fraction? Maybe the chain rule could be of any use, but I don't see how.

Differential equations notation
I've always wondered why does the differential equation notation for linear equations differ from the standard terminology of vector spaces.
We all know that the equation $y^{″}+p(x)y^{\prime }+q(x)y=g(x)$ for some function g is called linear and that the associated equation $y^{″}+p(x)y^{\prime }+q(x)y=0$ is called homogeneous. But why is that? WHY should mathematicians explicitly cause confusion with the rest of the theory of vector spaces?
What I mean by that is : Why not call the equation $y^{″}+p(x)y^{\prime }+q(x)y=g(x)$ an affine equation and call $y^{\prime }+p(x)y^{\prime }+q(x)y=0$ a linear equation? Because linear equations (in the sense of differential equations) are not linear in the sense of vector spaces unless they're homogeneous ; and linear equations (in the sense of differential equations) remind me more of a linear system of the form $Ax=b$ (which is called an affine equation in vector space theory) than of a linear equation at all.
Just so that I made myself clear ; I perfectly know the difference between linear equations in linear algebra and linear equations in differential equations theory ; I'm asking for some reason of "why the name".

Linear equation and linear differential equations
I remember noting from an algebra class that x and y of a linear equation neither divide or multiply with each other which is somewhat clear from the forms of linear equations:
General form of linear equation:
$Ax+By+C=0$
Slope intercept form:
$y=mx+b$
Is this also true for linear differential equations?
The definition goes like this: "A differential equation is said to be linear if the dependent variable and its differential coeficients (derivates) occur only in the first degree and not multiplied together."
$\frac{dy}{dx}=Py+Q$
Where P, Q are functions of x only. What exactly does this mean?
Does the algebraic linear equation has something to do with linear differential equation?