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The Separable differential equations exercise appears under the Differential equations Math section on Khan Academy. This exercise shows how to separate the $y$s from the $x$s on two different sides of the equation.

## Types of Problems Edit

There are six types of problems in this exercise:

1. Which of the following is the solution to the differential equation: The student is asked to find the solution to the differential equation using the initial condition.
2. Determine $a$ and $b$: The student is asked to determine the value(s) of a and b using the two equations and initial condition.
3. Determine $t$: The student is asked to determine the value(s) of $t$ using the derivative and properties given.
4. Determine the value of $y$: The student is asked to determine the value of $y$ given the value of $x$ and the graph of the differential equation.
5. Determine the value of $x$: The student is asked to determine the value of $x$ using the initial value and differential equation.
6. Which of the differential equations below are separable: The student is to determine which differential equations are separable by using factoring.

## StrategiesEdit

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

1. To solve a separate differential equation: Separate the x's (x and dx) and y's (y and dy), take the integral of each side, substitute the given coordinate to find the value of the constant, and then solve for y.
2. It is noted that not all differential equations are separable.
3. $\int{dx} = x$
4. $\int{dy} = y$

## 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.