Introduction to differential equations and initial value problems

The exercise appears under the Differential equations math section on Khan Academy. This exercise shows the basics of differential equations.
==Types of Problems==
There are six types of problems in this exercise:
# Find the sum of the two values: The student is asked to find the two values such that the first solution is a solution of the differential equation.
# Use substitution to match each differential equation on the left with an appropriate solution on the right: The student is asked to to match the differential equation with its solution.
# Which of the following is a general solution to the differential equation: The student is asked to determine the general solution to the differential equation than determine what C is for the particular solution with the initial condition.
# For what values of m and b is y=mx+b a solution: The student is asked to determine the values of m and b using the differential equation.
# Which of the following equations is the general solution to the differential equation: The student is asked to determine the general solution to the differential equation.
# Answer the following questions for the differential equation given the initial condition: The student is asked to determine the first and second derivative of the differential equation at the initial condition and use this information to determine the behavior of the graph.
==Strategies==
Knowledge of derivatives, factoring, and differential equations are encouraged to ensure success on this exercise.
# Since y=mx+b, then the derivative of y is equal is m
# Differential equations: solutions = function(s)
# You can find the value of m or \frac{dy}{dx} by grouping the x's together, setting the x factors equal to zero, and then plugging in that value into b.
# You can check your function solution by substituting it into the original \frac{dy}{dx} equation.
# When finding the sum of two values, factor using the least common multiple.
# y= f (x) = \frac{(d^2y)}{(dx^2)}
==Real-life Applications==
# Growth of a population: \frac{dy}{dt} = ky(L-y) when 0 < y(t) > L
# Chemical reaction conversion: \frac{dA}{dt} = kA^2
# Newton's second law of motion: m \frac{d^2x}{dt^2} = F
# Differential equations are used in heat transfer, electrical engineering, fluid mechanics, and modeling circuits.
Category:Math