1. Idea of a Solution of a Differential Equation
A differential equation does not ask for a single number as its solution. Instead, the goal is to find a function that satisfies the relationship between the variable and its derivatives. Any function that makes the differential equation true is called a solution.
Because derivatives remove constants, solving a differential equation usually introduces arbitrary constants back into the expression. This naturally leads to two types of solutions: general solutions and particular solutions.
1.1. Why solutions come in families
Different functions can have the same derivative behaviour. This is why differential equations often lead to a family of solutions parameterized by one or more arbitrary constants. Extra information, if provided, narrows the family to a single specific solution.
2. General Solution
A general solution of a differential equation is a solution that contains as many arbitrary constants as the order of the equation.
These constants represent the fact that differentiating removes information, and solving brings those missing constants back. A general solution captures all possible solutions of the equation.
2.1. Properties of a general solution
- Contains one arbitrary constant for a first-order equation.
- Contains two arbitrary constants for a second-order equation, and so on.
- Represents a whole family of curves.
- Any particular member of this family can satisfy additional conditions if needed.
2.2. Example of a general solution
Consider the differential equation:
\(\frac{dy}{dx} = 3x^2.\)
Integrating gives:
\(y = x^3 + C,\)
where \(C\) is an arbitrary constant. This expression represents the general solution, because it includes all possible functions whose derivative is \(3x^2\).
3. Particular Solution
A particular solution is obtained by giving specific values to the arbitrary constants in the general solution. This happens when extra information such as an initial value or a boundary value is provided.
Such extra information selects one curve out of the entire family of solutions.
3.1. How a particular solution is formed
Suppose the general solution has the form
\(y = x^3 + C.\)
If we are told that \(y = 9\) when \(x = 2\), substitute these values:
\(9 = 2^3 + C \Rightarrow C = 1.\)
Thus the particular solution becomes:
\(y = x^3 + 1.\)
3.2. Example with a second-order differential equation
Take the equation
\(\frac{d^2y}{dx^2} = 0.\)
Integrating twice gives the general solution:
\(y = C_1 x + C_2.\)
If the extra conditions are \(y(0) = 3\) and \(y(1) = 5\), substitute these values to find \(C_1\) and \(C_2\). The unique result is the particular solution that fits both conditions.
4. General vs Particular Solutions
Both solutions are connected, but they serve different purposes:
- The general solution represents all possible behaviours allowed by the differential equation.
- The particular solution is a single curve chosen to satisfy additional conditions.
- Extra conditions are essential to select a unique solution from a family.
4.1. Visual idea
Imagine a bundle of curves representing the general solution. A particular solution picks exactly one of these curves by passing through specific given points.
5. Worked Examples
Here are a few small examples showing how general and particular solutions arise naturally when solving differential equations.
5.1. Example 1: General → Particular
Solve \(\frac{dy}{dx} = 2y\).
Integrate:
\(\frac{dy}{y} = 2 dx \Rightarrow \ln|y| = 2x + C.\)
Exponentiate:
\(y = Ae^{2x},\)
where \(A\) is an arbitrary constant. This is the general solution.
If given that \(y = 6\) at \(x = 0\):
\(6 = A e^0 \Rightarrow A = 6.\)
Particular solution:
\(y = 6e^{2x}.\)
5.2. Example 2: Two arbitrary constants
Consider the equation
\(\frac{d^2y}{dx^2} + 4y = 0.\)
Its general solution is:
\(y = C_1 \cos 2x + C_2 \sin 2x.\)
Two arbitrary constants appear because it is a second-order equation. Extra conditions (such as initial or boundary values) determine \(C_1\) and \(C_2\) and lead to a particular solution.
6. Notes and Observations
Some quick points that are helpful to keep in mind:
- The number of arbitrary constants equals the order of the differential equation.
- A general solution forms a whole family of curves.
- A particular solution satisfies given conditions and is unique for those conditions.
- Initial conditions or boundary conditions are used to determine the constants.