These are my complete notes for Separable Equations & Applications in Integral Calculus.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@berkeley.edu.
Summary of Separable Equations & Applications (Integral Calculus)
?. Separable Equations & Applications.
# Separable Differentiable Equations: A differential equation of the form (dy/dx) = f(y)g(x) is called separable. We separate the variables using the form (1 / f(y)) dy = g(x)dx.
The solution is found by antiderivating both sides.
# How To Solve All Separable Equations:
Step 1: Get all "y" terms on one side.
Step 2: Get the "dx" on both sides.
Step 3: Integrate both sides in respect to each variable.
Step 4: Solve for "c" if possible.
# Exponential Change: Exponential Change occurs when the rate of change is proportional to the amount present. This is when you can multiply the amount by a factor each time to get a new amount:
y = k * x
(dy/dt) = k * y
# Compound Interest:
$$A(t) = A_0 (1+\frac{r}{k})^{kt}$$ A0 = amount invested (initial condition).
r = interest rate (decimal rate).
k = # of times per year compounded.
t = time in years (or whichever time is specified by the question)
# Modeling Growth with Other Bases: $$y = y_0 b^{\frac{-t}{n}}$$ y0b = the desired growth.
n = the amount of time until the value is multiplied by b, i.e. the length of one growth cycle.
t = time in years (or whichever time is specified by the question)
#
Rule .
For the most distilled understanding of separable equations possible, A is just going to represent moving x and y to either side any way you can. If there is no x or y, pretend it exists. For EXPONENTIAL CHANGE, the equation is:
$$y = y_0 e^{rt}$$
, where y is the total after time t, r is the rate, and y0 is the initial amount. To determine any equation, you just need to know y (generally always either y0 doubled or in 30 years), and two of either r, t, or y0. The irregular compound is the (1 + r/k) one described in the "Compound Interest" Blue Section, where r is always a decimal.
The half-life form is $$y = y_0 e^{-kt}$$ , with the half-life itself being ½y0.
The half-life form is $$y = y_0 e^{-kt}$$ , with the half-life itself being ½y0.
# Trigonometric Antiderivatives:
$$\int \sec(x) dx = \ln(|\sec(x) + \tan(x)) + c$$ $$\int \csc(x) dx = \ln(|\csc(x) - \cot(x)|) + c$$ $$\int \tan(x) dx = \ln(|\sec(x)|) + c$$ $$\int \cot(x) dx = \ln(|\sin(x)|) + c$$ $$\int \tan^2(x) dx = -x + \tan(x) + c$$ $$\int \cot^2(x) dx = -x-\cot(x)+c$$
