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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.