An equation is separable when dy/dx can be written as a function of x times a function of y. The two variables can then be pulled apart.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The method
Move everything with y to one side and everything with x to the other, then integrate both sides.
One constant, not two
Integrating both sides produces two constants, but only their difference matters. Write a single C.
Implicit answers
The result may not be solvable for y. An implicit relation is still a complete answer.
Using an initial condition
Substitute the starting values to find C. That turns the family into one particular solution.
Watch for lost solutions
Dividing by a function of y can quietly discard the constant solution where that function is zero. Check the equilibria separately.
Separate, then integrate
Rearrange so all the y terms sit with dy and all the x terms with dx, then integrate both sides. The method applies only when the equation separates, which is checked before anything else.
One constant, not two
Integrating both sides produces a constant on each, but their difference is a single arbitrary constant. Writing one C is correct rather than a shortcut.
The answer may be implicit
Sometimes y cannot be isolated after integrating. An implicit solution is a legitimate answer, and forcing an explicit form can introduce sign errors or lose branches.
Dividing can lose solutions
Separating usually involves dividing by a function of y, which is invalid where that function is zero. Those values are often equilibrium solutions, and they must be checked separately and reinstated.
Step 2: Try It Yourself
Tap and try it out.
- The equationdy/dx = a·y
- Through(0, 1)
Every short line shows the slope the equation demands at that point. The curve simply follows them, and the marked point is the initial condition that picks it out of the family.
Step 3: Watch an Example
One step at a time.
Watch Kai Separate an Equation
Kai solves dy/dx = xy with y = 2 when x = 0.
- Step 1
He divides both sides by y and multiplies by dx, giving dy/y = x dx.
Step 4: Your Turn
Practice makes it stick.
The Constant
Problem 1 of 2
y = Ae^(2x) with y = 6 when x = 0. What is A?
The Lost One
Problem 2 of 2
dy/dx = xy. Dividing by y risks losing which constant solution?
Pull Them Apart
1 of 8
y = Ae^(3x) with y = 5 at x = 0. What is A?
2 of 8
y = 4e^(2x). What is y at x = 0?
3 of 8
Is dy/dx = xy separable? 1 yes, 0 no.
4 of 8
Is dy/dx = x + y separable? 1 yes, 0 no.
5 of 8
How many arbitrary constants does the general solution of a separable first-order equation have?
6 of 8
dy/dx = 3y with y = 2 at x = 0. What is y at x = 0 still?
7 of 8
Order the steps of separation of variables.
- 1Move all y terms one side and all x terms the other
- 2Integrate both sides and add one constant
- 3Use the initial condition to find that constant
- 4Check the equation is separable
8 of 8
Is dy/dx = y² separable? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
y = Ae^(5x) with y = 9 at x = 0. What is A?
Question 2 of 2
What makes an equation separable?
What You Learned
- A separable equation splits into a function of x times a function of y.
- Move each variable to its own side, integrate, and keep a single constant.
- Dividing by a function of y can lose the constant solution where it vanishes.