Separable differential equation
An equation for dy/dx that can be rearranged to have all the y terms on one side and all the x terms on the other. Then integrate both sides.
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Analogy
Sorting laundry: all the y-items into one pile, all the x-items into the other.
Example
Separate: dy/y = 2x dx.Integrate: ln y = x^2 + C.So y = A e^(x^2).y(0) = 3 gives A = 3, so y = 3e^(x^2).
In practice
Models growth, cooling and decay, where the rate depends on the current amount.
Practice
Flashcards
- What is the solution of dy/dx = 2xy with y(0) = 3?y = 3e^(x^2).
Questions people ask
What is a differential equation?
It is an equation that links a function to its derivatives. Instead of asking for a number, you must find a whole function. For dy/dx = 2y, the solutions are y equals A times e to the 2x, where A depends on the starting condition.
What is an initial condition for?
The general solution contains a constant, so there are infinitely many solutions. An initial condition, such as y(0) = 3, picks the one that passes through a given point. It lets you work out the value of the constant. In the example, y(0) = 3 forces A to be 3, so only one curve remains.