What is a Solution?
A solution of a differential equation is a function $y = f(x)$ that satisfies the equation — meaning when $y$ and its derivatives are substituted, the equation becomes an identity.
Show that $y = e^{2x}$ is a solution of $\frac{dy}{dx} - 2y = 0$
Given $y = e^{2x}$, then $\frac{dy}{dx} = 2e^{2x}$.
Substituting: $2e^{2x} - 2 \cdot e^{2x} = 0$ ✓
General Solution
The solution which contains arbitrary constants, equal in number to the order of the DE, is called the general solution (or complete primitive).
The general solution represents a family of curves. Each value of the arbitrary constants gives a different member of this family.
Examples
| DE (Order) | General Solution | # of Constants |
|---|---|---|
| $\frac{dy}{dx} = 2x$ (order 1) | $y = x^2 + C$ | 1 |
| $\frac{d^2y}{dx^2} + y = 0$ (order 2) | $y = A\cos x + B\sin x$ | 2 |
| $\frac{d^2y}{dx^2} = 0$ (order 2) | $y = Ax + B$ | 2 |
where $C_1, C_2, \ldots, C_n$ are arbitrary constants
Particular Solution
A solution obtained from the general solution by giving specific values to the arbitrary constants is called a particular solution. It satisfies given initial or boundary conditions.
Find the particular solution of $\frac{dy}{dx} = 2x$ given $y(0) = 3$
- Integrate: $y = x^2 + C$ — this is the general solution.
- Apply initial condition: $y = 3$ when $x = 0$: $\quad 3 = 0 + C \Rightarrow C = 3$
- Particular solution: $y = x^2 + 3$
Find the particular solution of $\frac{d^2y}{dx^2} + y = 0$ given $y(0) = 1,\ y'(0) = 0$
General solution: $y = A\cos x + B\sin x$
At $x=0$: $y = A = 1 \Rightarrow A = 1$
$y' = -A\sin x + B\cos x$; at $x=0$: $y'= B = 0 \Rightarrow B = 0$
Particular solution: $y = \cos x$
Singular Solution
Some DEs have solutions that are not obtainable from the general solution for any value of the arbitrary constant. These are called singular solutions.
🔑 Key Takeaways
- A solution of a DE is a function that satisfies it identically.
- General solution: contains arbitrary constants (= order of DE).
- Particular solution: obtained by assigning values to constants using initial conditions.
- An $n$th order DE has $n$ arbitrary constants in its general solution.
- The general solution represents a family of curves; particular solution is one specific curve.