Calculus l - Differential Equations

Differential equations: Equations that involve derivatives. The solution to a differential equation is not a set of values, but a function or set of functions. 

Methods to solve:

Separation of Variables method: Only works if variables are isolatable. 

Step 1.) Get all like variables on the same side. For example, all the x’s on one side, and all the y’s on the other. 

Step 2.) Integrate! (After you integrate, you’ll get a “+ C” on both sides, just join the “+ C”’s into one “+ C” on the side of the independent variable (the variable you are integrating with respect to, most commonly known as x)).

Step 4.) Isolate the dependent variable if not yet done (most commonly known as y).

Step 5.) (This step depends, you can also do this step before step 4) If you are given an initial condition, say y(1) = 2, then you can plug in 1 for x and 2 for y to solve for c. This will help us find a general equation. 

To check if a function satisfies the differential equation, we can plug it in for “y” and check. 

Tips: 

Whenever we have something multiplied by/divided by/raised to the base of/ exponent raised to the base of our constant “C”, we just convert that entire term into a “+ C”. If something is multiplied to a constant it is still just some constant.

 

If we have more than one “+ C”, then we can combine them into one “+ C” and just put it on one side.

To cancel a natural log, we can raise both sides to a base e. To cancel an absolute value, we can just write the plus or minus value of the other side. 

Growth V.S. Decay

Growth differential equation:

The actual function used to find solutions (found by separation of variables method on growth differential):

r is just k. 

Decay differential equation:

r is just k.

Actual function used to find solutions (found by separation of variables method on decay differential):

,  k is just the negative natural log of the percent decay in decimal over the time it takes for that percentage of decay. 

Newton’s Law of Cooling

Used to answer “How long until you get the desired temperature?”. Questions relating time and temperature. 

Differential equation:

*For Newton's law of cooling, if k is not given, assume a good approximation would be 0.0005.*

Slope fields: a bunch of small dashes on points that exist on a graph that represent the slope of the function at certain points. These dashes represent the derivative of a point on the graph, but extend outwards both sides to form an actual dash or line. They are used to approximate the slope with the help of delta x. Delta x will tell the distance from one x-value to another, to approximate the slope you would connect following the path of the dashes that exist at these x-values on the graph. To better the approximation, make delta x a smaller number. 

Linearization

Local Linear Approximation or “Linearization” of f(x) at the point x = a.

y = f(a) + f’(a)(x-a)

Newton’s Method (may not work in all cases, for example, if you pick an x-value that has a y min or max, then the tangent line will have a slope of zero which is just a horizontal line that will never intersect the x-axis again and hence, won’t get you the next x-value for approximation). 

Steps to solve:

Step 1.) Get a rough sketch of the graph and identify the x-value that’s closest to the zero. Avoid picking an x-value that you know will give a max or min y-value. The x-value you pick will be your x1, (A.K.A your first guess).

Step 2.) Using the formula above, original function, function’s derivative, and x1, you can plug in x1 for xn and solve for xn+1 or x1+1 which is just x2. Repeat the process, each time replacing the xn with the value you obtained from the previous calculation, and incrementing n up by 1. For example, in the next incrementation, you would plug in the x2 that you found using the formula the first time for xn and solve for x2+1 or x3. 

Step 3.) Depending on “to __ places of accuracy”, (say, for example, 4 places of accuracy), you would keep repeating the process until you notice that the value you obtained from the previous calculation has the exact same number(s) in the same place value position as the calculation you just did.

For example x3 = 2.35443 and x4 = 2.30284. Notice how x3 and x4 seem to settle on the first two values being 2.3. Continue this process until you know for sure up to, in this example, 4 decimal places. 

The Trapezoid Rule is essentially the average of the left and right areas: 

b and a are the bounds, n is the number of intervals.

The Simpson’s Rule (most accurate approximation):

b and a are the bounds, n is the number of intervals.