7.7 Finding Particular Solutions Using Initial Conditions and Separation of Variables

Separation of variables

To solve a separable differential equation, get all y-terms (with dy) on one side and all x-terms (with dx) on the other. Then integrate both sides: ∫g(y)dy = ∫f(x)dx. Add a constant of integration +C (only one is needed, since both sides produce constants that combine).

Using initial conditions

A general solution contains an arbitrary constant C, representing a family of curves. Plugging in an initial condition (a given point like y(0)=3) lets you solve for the exact value of C, producing the particular solution that passes through that specific point.

Solving for y explicitly

After integrating and solving for C, isolate y as a function of x when possible. Watch for absolute values from ln|y| — you may need to consider the sign of y based on the initial condition to remove the absolute value correctly and get an explicit formula.

Key terms

Separable differential equation

A differential equation that can be written so all y-terms (with dy) are on one side and all x-terms (with dx) are on the other.

General solution

A solution to a differential equation containing 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 in the general solution.

Initial condition

A given point (x₀, y₀) that the solution curve must pass through, used to determine the value of C.

Steps to solve using separation of variables

1) Separate variables (y-terms with dy, x-terms with dx). 2) Integrate both sides, add +C. 3) Use the initial condition to solve for C. 4) Solve for y explicitly if possible.

Try an AP-style question

Solve dy/dx = 2x with y(0) = 3.

1. y = x² − 3
2. y = x² + 3x
3. y = 2x + 3
4. y = x² + 3