Separable Differential Equations
Definition and Solution of a Separable Differential
We have seen how to check if a given solution is a solution to a differential equation. Now we will begin our quest to find the solution. Although most differential equations are unsolvable, there are some that do have a solution that can be found. We begin with the simplest type.
A differential equation is called separable if it can be written as
Solve the following differential equations
Finding a Particular Solution
Solve the differential equation
dy x + 1
Notice that this is separable. We separate the variables.
y dy = (x + 1) dx
Now integrate both sides to get
Now use the initial condition. That is, plug in 0 for x and 4 for y.
C = 8
We can write
Multiply by 2 to get
y2 = x2 + 2x + 16
Finally take a square root to obtain
Find the equation of the function that has the following properties:
Slope = y sin x
and passes through the point (0,2).
Since the slope is the derivative, we have
dy / dx = y sin x
Integrating both sides gives
ln y = sin x + C
Since it passes through (0,2), we plug in 0 for x and 2 for y.
ln(2) = sin(0) + C
ln(2) = C
ln y = sin x + ln(2)
y = esin x + ln(2) = esin x eln(2)
y = 2esin x