Math · Calculus II · Concept
Separable differential equations
A differential equation relates a function to its derivatives, and solving it means finding the function. A first-order equation is separable when it can be written dy/dx = g(x)h(y), and it is solved by separation of variables: divide by h(y), multiply by dx, and integrate each side with respect to its own variable. The result, with one constant of integration, is the general solution; an initial condition such as y(0) = 3 picks out one particular solution.
What a solution is
A solution of a differential equation is a function that makes the equation true for every x in an interval. To check a proposed solution, differentiate it and substitute: y = 3e^(x²) solves dy/dx = 2xy because its derivative, 6xe^(x²), is 2x times y.
Separate the variables
If dy/dx = g(x)h(y), rewrite it as dy/h(y) = g(x) dx, with every y on the left and every x on the right. Then integrate both sides; one constant C, on one side, is enough.
Use the initial condition
The general solution contains C. Substitute the initial condition to find C; that gives the particular solution through the given point.
Watch for lost solutions
Dividing by h(y) assumes h(y) ≠ 0. Each constant y = c with h(c) = 0 is also a solution, an equilibrium, and the division can lose it. For dy/dx = 2xy, y = 0 is such a solution.
Growth, decay and cooling
dy/dt = ky separates to y = y₀e^(kt), the exponential model. Newton’s law of cooling, dT/dt = −k(T − Tₛ), separates the same way to T = Tₛ + (T₀ − Tₛ)e^(−kt): the difference from the surroundings’ temperature Tₛ decays exponentially.
Common mistakes
- Integrating only one side, or leaving a y on the x side.
- Adding a separate constant to each side: the two combine into one.
- Dropping the absolute value in ln|y|; then e^C becomes a constant A that may be positive or negative.
- Losing an equilibrium solution such as y = 0 when dividing by h(y).
Key terms
- Differential equation
- An equation involving an unknown function and its derivatives, such as dy/dt = ky. An initial value, such as y(0) = 5, picks out one particular solution.
- Separable differential equation
- A first-order differential equation that can be written dy/dx = g(x)h(y), a function of x times a function of y. Moving every y to one side and every x to the other, then integrating both sides, solves it.
- Initial-value problem
- A differential equation together with a condition that picks out one solution, such as f′(x) = 3x² + 2 with f(1) = 5. For an antiderivative, the condition determines the constant C.
- Constant of integration
- The + C added to an antiderivative, because the derivative of any constant is 0. On a domain split into separate pieces, each piece can have its own constant.
- Natural logarithm
- The logarithm with base e, written ln x. It undoes eˣ: ln(eˣ) = x for every x, and e^(ln x) = x for x > 0.
- Newton’s law of cooling
- A model in which an object’s temperature changes at a rate proportional to the difference between its temperature and its surroundings’: dT/dt = −k(T − Tₛ). Its solution, T = Tₛ + (T₀ − Tₛ)e^(−kt), approaches the surrounding temperature exponentially.
Work through an example
Solve dy/dx = 2xy with the initial condition y(0) = 3.
Solve a separable differential equation →Sources and scope
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