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Introduction to differential equations and initial value problems(1)
Introduction to differential equations and initial value problems(2)

The Introduction to differential equations and initial value problems exercise appears under the Differential equations math section on Khan Academy. This exercise shows the basics of differential equations.

Types of ProblemsEdit

There are six types of problems in this exercise:

  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.

StrategiesEdit

Knowledge of derivatives, factoring, and differential equations are encouraged to ensure success on this exercise.

  1. Since y=mx+b, then the derivative of y is equal is m
  2. Differential equations: solutions = function(s)
  3. 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.
  4. You can check your function solution by substituting it into the original \frac{dy}{dx} equation.
  5. When finding the sum of two values, factor using the least common multiple.
  6.  y''= f'' (x) = \frac{(d^2y)}{(dx^2)}

Real-life ApplicationsEdit

  1. Growth of a population: \frac{dy}{dt} = ky(L-y) when 0 < y(t) > L
  2. Chemical reaction conversion: \frac{dA}{dt} = kA^2
  3. Newton's second law of motion: m \frac{d^2x}{dt^2} = F
  4. Differential equations are used in heat transfer, electrical engineering, fluid mechanics, and modeling circuits.
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