Solve System of Differential Equations with Expert Help

Solve the differential equation:
${\displaystyle y^{\prime }\left(x\right)=y\left(x\right)-x-1+\frac{1}{x-y\left(x\right)+2}}$
with initial condition ${\displaystyle y\left(0\right)=0}$

A question asks us to solve the differential equation
${\displaystyle -u\left(x\right)=\delta \left(x\right)}$
with boundary conditions
${\displaystyle u(-2)=0\text{and}u\left(3\right)=0}$ where ${\displaystyle \delta \left(x\right)}$ is the Dirac delta function. But inside the same question, teacher gives the solution in two pieces as ${\displaystyle u=A(x+2)\text{for}x\le 0\text{and}u=B(x-3)\text{for}x\ge 0}$. I understand when we integrate the delta function twice the result is the ramp function R(x). However elsewhere in his lecture the teacher had given the general solution of that DE as
${\displaystyle u\left(x\right)=-R\left(x\right)+C+Dx}$

I was toying with equations of the type ${\displaystyle f(x+\alpha )=f^{\prime }\left(x\right)}$ where f is a real function. For example if ${\displaystyle \alpha =\frac{\pi }{2}}$ then the solutions include the function ${\displaystyle f_{\lambda ,\mu }\left(x\right)=\lambda \cos (x+\mu )}$. Are there more solutions?
On the other hand, if I want to solve the equation for any ${\displaystyle \alpha }$, I can assume a solution of the form ${\displaystyle f\left(x\right)=e^{\lambda x}}$, and find ${\displaystyle \lambda }$ as a complex number that enables me to solve the equation...
I was wondering: is the set of the solutions of dimension 2 because the derivative operator creates one dimension and the operator ${\displaystyle \varphi :f\left(x\right)\to f(x+1)}$ adds another one? Is there some litterature about this kind of equations?

${\displaystyle \frac{dy}{dt}+2y=1;y\left(0\right)=\frac{5}{2}}$
when i solve this i get ${\displaystyle \ln (-4)=c}$ now the problem is how to solve ${\displaystyle \ln (-4)}$?

I have heard about a lot of fancy numerical methods for solving ODEs. I know that there are methods that (assuming sufficient smoothness) asymptotically give a low error, like the Runge-Kutta methods. These estimate the solution in a set of points ${\displaystyle t_0,t_1}$, etc. But what if I want to have a function that is close to the correct solution everywhere, not just in a discrete set of points?
I can extend the numerical solution to a piecewise linear function. This will be a continuous function and it will converge to the correct solution if the step-size goes to zero.
But the error estimate will be poor in most places unless I use a very small step-size, which rather defeats the purpose of using a high-order method. So how does one go about estimating the solution in practice between the ${\displaystyle t_i}$?

f(t)=sin(2t)-t cos(t)

${\displaystyle (y-x{y}^2)dx+(x+x^2{y}^2)dy=0;\text{when}y\left(1\right)=1}$

Identify the following linear differential equation.
${\displaystyle x^2{y}^{\prime }+y^2=0}$

Brine containing 1.5lb/gal of salt enters a tank at the rate of 2gal/min and the mixture well stirred leaves at the same rate. The tank initially contains 100 gal of brine with10 lbs of dissolved salt. When will the amount of salt in the tank be 20 lbs?

${\displaystyle \frac{dy}{dx}+5y=15}$

Radium decomposes at a rate proportional to the amount present. In 100 years, 100 mg of radium decompose to 96mg. How many mg will be left after another 100 years What is the "half life" (the time required to decompose half the initial amount) of radium?

Non-Homogeneous Linear Differential Equation (The Method of Undetermined Coefficients).
Solve the equation ${\displaystyle (D+1)y=\sin x}$

${\displaystyle y^{\prime }+\frac{1}{x}y=\cos \left(x\right)}$

${\displaystyle xydx+\left\{\frac{1}{2}{x}^2(1-y\ln y-y)\right\}dy=0}$

Solve for G.S. / P.S. for the following differential equations using the method of solution for homogeneous equations.
${\displaystyle -(3x+25y)dx+(25x+3y)dy=0}$

When a body is removed from an oven, its temperature is measured at ${\displaystyle {230}^{\circ }F}$. Four minutes later its temperature is ${\displaystyle {120}^{\circ }F}$. If the room temperature is found to be ${\displaystyle {70}^{\circ }F}$, how long will it take for the chicken to cool off to a temperature of ${\displaystyle {90}^{\circ }F}$. What is the steady state temperature of the body?

Solve for the general solution of the differential equation ${\displaystyle (x+2y-1)dx+(2x+4y-3)dy=0}$ using Substitution Suggested by the Equation.

${\displaystyle \frac{dy}{dx}=-\frac{ax+by}{bx+cy}}$

Find the differential equations of the following. Show complete solution. ${\displaystyle \frac{dy}{dx}+y=e^{2x}}$