7.6 Finding General Solutions Using Separation of Variables

What is separation of variables?

Separation of variables solves a differential equation by getting all y terms (with dy) on one side and all x terms (with dx) on the other. Then integrate both sides separately. This works when dy/dx can be written as a product/quotient of a function of x and a function of y: dy/dx = f(x)g(y).

Steps to solve

1) Rewrite dy/dx = f(x)g(y). 2) Divide by g(y): dy/g(y) = f(x)dx. 3) Integrate both sides, adding a single constant +C. 4) Solve for y if possible to get the general solution. Don't forget: this produces a family of curves, one for each value of C.

From general to particular solution

A general solution contains an arbitrary constant C. Using an initial condition (a given point the solution passes through), substitute values to solve for C, producing the unique particular solution. Always check domain restrictions (e.g., avoid dividing by zero) when separating variables.

Key terms

Separation of variables

A method to solve dy/dx = f(x)g(y) by rewriting as dy/g(y) = f(x)dx and integrating both sides.

General solution

The solution to a differential equation that includes an arbitrary constant C, representing a family of curves.

Particular solution

A specific solution obtained by using an initial condition to solve for the constant C.

Initial condition

A given point (x0, y0) that the solution must satisfy, used to find the value of C.

Key caution when separating variables

Check for values where g(y) = 0 before dividing, since dividing by zero is undefined and may hide constant solutions.

Try an AP-style question

What is the general solution of dy/dx = xy?

1. y = x²/2 + C
2. y = Cx
3. y = Ce^x
4. y = Ce^(x²/2)