What does it mean for energy to be conserved in a system and why is this important?
Energy being conserved with in a system means that no energy is lost due to motion or collisions within the system. All the energy is simply transferred between objects or from one type to another. For example, let's say a ball is held in the air above ground at rest. This means that it has no Kinetic Energy but it does have Gravitational Potential Energy. Then the ball is dropped and just before it hits the ground it will have only Kinetic Energy and no Gravitational potential energy. This is important because it will not only make calculations simpler, but also allows us to understand that no energy can be created or lost randomly.
How do you find the general solution of a differential equation? (Consider the cases where the differential equation = f(x) and the differential equation = 0)
The general solution of a differential equation is equal to the sum of it's solution to the particular and it's solution to the homogeneous. The homogeneous solution is found by setting the differential equation equal to 0 and solving as you normally would. (This can usually be solved using separation of variables). The particular solution is found by letting y(x) take the form of f(x) with constant coefficients and then solving the differential equation. For example if f(x) = e^x then we guess that y(x) = Ae^x. Then we can plug in y(x), y'(x), etc. to verify we get the correct solution and, in the case that we have initial conditions, find the constant A. Note, that if the differential equation is purely homogeneous then the particular part of the general solution is 0. Where as if the differential equation = f(x) then we must also consider the solution to the homogeneous of that equation to find the general solution.
What steps should someone take to find the integral of a general function? (Step examples: Simplify the expression, attempt substitution, etc.)
There are various strategies to solve any integral. Some integrals are definitely easier to compute than others, but there are still some steps that should always be taken: 1) Simplify the integrand as much as possible 2) If it is a definite integral, check if the function is continuous within its bounds. Else break it up into multiple integrals. This is especially applicable to piece-wise functions 3) Check if it's similar to a function you already know how to integrate, this might save you time and effort. If it is similar, substitute if possible to get to an easier integral. After doing the above the rest comes down to personal preference when solving the problem. Most integrals will be able to be solved with a substitution or with trigonometric identities. However, if those don't work look to apply partial fractions or integration by parts. Sometimes you may need to do a combination of these in order to get the answer. The best way to figure it out and get better is practice!