Separable Differential Equations
Definition and Solution of a Separable Differential
Equation 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
Exercises:
Solve the following differential equations
Finding a Particular Solution Example Solve the differential equation
dy x + 1
Solution Notice that this is separable. We separate the variables. y dy = (x + 1) dx Now integrate both sides to get
y2
x2 Now use the initial condition. That is, plug in 0 for x and 4 for y.
42
02 So that C = 8 We can write
y2
x2 Multiply by 2 to get y2 = x2 + 2x + 16 Finally take a square root to obtain
Example Find the equation of the function that has the following properties: Slope = y sin x and passes through the point (0,2).
Solution Since the slope is the derivative, we have dy / dx = y sin x Separating gives
dy 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 We have ln y = sin x + ln(2) exponentiation gives y = esin x + ln(2) = esin x eln(2) Hence y = 2esin x
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