Differential+Equations

Differential Equations Intro
Basics: 1. First convert all notation in terms of x, y, dx, and dy. 2. If possible separate all variables to each side. ie (all x's to one side and all y's to the other side) 3. Integrate both sides of the equation, combining all constants to form one constant. 4. Plug in given points to find the particular solution of an equation.

Example: Find the particular solution for (0,5) of math y'=yx^2 math math \frac{dy}{dx}=yx^2 math math \frac{dy}{y}=x^2dx math math ln|y|=\frac{x^3}3+C math math y=e^{\frac{x^3}3+C}=C*e^{\frac{x^3}3} math math 5=C*e^0 math math C=5 math math y=5*e^{\frac{x^3}3} math

Rate of Change in Inversely or Directly Proportional to Value:

// Q // = Thermal energy in [|joules] // h // = [|Heat transfer coefficient] // A // = Surface area of the heat being transferred // T // = Temperature of the object's surface and interior (since these are the same in this approximation) // T // env = Temperature of the environment Δ // T // ( // t // ) = // T // ( // t // ) − // T // env is the time-dependent thermal gradient between environment and object
 * Newton's law of cooling-** //the rate of heat loss of a body is proportional to the difference in temperatures between the body and its surroundings//