General Solution — Definition, Formula & Examples
A general solution is the complete family of all solutions to a differential equation, expressed using one or more arbitrary constants. Each specific value of the constant(s) picks out one particular solution from this family.
The general solution of an th-order ordinary differential equation is a solution containing independent arbitrary constants such that every solution of the equation (under standard existence and uniqueness conditions) can be obtained by assigning specific values to these constants.
How It Works
When you solve a differential equation by integration, each integration step introduces an arbitrary constant. These constants remain unspecified until you apply additional information such as initial conditions or boundary conditions. The general solution with its free constants represents an entire family of curves. Substituting specific values for the constants yields a particular solution that satisfies the given conditions.
Worked Example
Problem: Find the general solution of the differential equation dy/dx = 6x.
Integrate both sides: Integrate with respect to x to reverse the derivative.
Identify the general solution: The result contains one arbitrary constant C, which is expected for a first-order equation. This expression represents the entire family of parabolas that satisfy the equation.
Example of a particular solution: If an initial condition states y(0) = 5, substitute x = 0 and y = 5 to find C = 5, giving the particular solution.
Answer: The general solution is , where is an arbitrary constant.
Why It Matters
In modeling physical systems — population growth, cooling processes, circuit behavior — the general solution captures every possible scenario. Applying measured data as initial or boundary conditions narrows it to the one particular solution that matches reality.
Common Mistakes
Mistake: Forgetting the arbitrary constant after integration and writing a single function as the 'general solution.'
Correction: Each integration introduces a constant. An th-order ODE must have arbitrary constants in its general solution. Without them, you have only one particular solution, not the full family.