A separable differential equation is a first order equation where the variables can be rearranged so all y terms are on one side and all x terms are on the other. This form matters because it turns a differential equation into two ordinary integrals. Many growth, decay, cooling, and motion models become solvable once you recognize that the variables can be separated.
The main goal is to find a function y(x) that satisfies both the equation and any starting condition.
Understanding Calculus: Separable Differential Equations
The key idea is to treat a rate of change as a relationship between tiny changes. When the rate depends on a factor involving the input and a factor involving the output, each factor can be moved beside its own tiny change. This works because multiplication can be rearranged, but division needs care.
If you divide by an expression containing y, first check when that expression could be zero. A solution with y equal to zero may be lost during division.
It must be tested separately in the original equation. These lost solutions are often called equilibrium solutions because the output stays at one value over time.
Integration introduces a constant because many different functions have the same derivative. It is best to use one constant after both sides have been integrated, rather than placing a different constant on each side. An initial condition selects one member of this family of solutions.
In a physical model, that condition is a measurement at a particular time or position. For example, a population model needs a known starting population.
A cooling model needs the starting temperature. Without that information, the equation describes possible behavior but not one definite situation.
Logarithms appear often in these problems. When the integral includes one divided by y, its antiderivative is the natural logarithm of the absolute value of y. The absolute value matters because y can be positive or negative, while a logarithm only accepts positive inputs.
Later steps may involve exponentiating both sides. At that stage, the constant can be rewritten because an arbitrary positive multiplier represents the exponential of an arbitrary constant.
Students should not assume this multiplier is positive unless the derivation requires it. The sign may need its own constant, depending on the interval being studied.
A final answer should be checked, not just simplified. Differentiate the proposed function and substitute it back into the original rate equation. This catches missing factors, incorrect powers, and sign errors.
Pay attention to the interval where the solution is valid. A formula involving a logarithm cannot cross zero without special treatment, and a denominator cannot become zero. Real models make these limits meaningful.
A population cannot usually be negative, a concentration may be restricted to nonnegative values, and time may begin at zero. The calculus gives a formula, but the context tells you which part of that formula describes the actual system.
Key Facts
- A separable equation can be written as dy/dx = f(x)g(y).
- Separate variables by rewriting as 1/g(y) dy = f(x) dx, when g(y) is not zero.
- Integrate both sides: ∫ 1/g(y) dy = ∫ f(x) dx + C.
- Use an initial condition y(x0) = y0 to solve for the constant C.
- Example: dy/dx = 3x y gives ∫ 1/y dy = ∫ 3x dx, so ln|y| = 3x^2/2 + C.
- After integrating, solve for y when possible, but an implicit solution is sometimes acceptable.
Vocabulary
- Differential equation
- An equation that contains a function and one or more of its derivatives.
- Separable equation
- A first order differential equation that can be rearranged so each variable appears with its own differential.
- General solution
- A family of solutions that includes an arbitrary constant.
- Initial condition
- A given value such as y(0) = 2 that selects one specific solution from the general family.
- Implicit solution
- A solution where x and y remain in one equation instead of being solved explicitly for y.
Common Mistakes to Avoid
- Forgetting the dx or dy after separating variables is wrong because the differentials show which variable each integral uses.
- Separating when the equation is not actually separable is wrong because not every expression dy/dx = F(x,y) can be written as f(x)g(y).
- Dropping the constant of integration is wrong because the differential equation usually has infinitely many solution curves before an initial condition is applied.
- Dividing by a factor such as y without checking y = 0 is wrong because it may remove a valid constant solution.
Practice Questions
- 1 Solve the separable differential equation dy/dx = 4x y with initial condition y(0) = 3.
- 2 Solve dy/dx = x/(y) with initial condition y(0) = 2, assuming y is positive.
- 3 Explain why dy/dx = x + y is not separable in the simple form dy/dx = f(x)g(y), and describe what feature prevents direct separation.