General and Particular Solutions of a Differential Equation

9.3 General and Particular Solutions of a Differential Equation

From Algebraic Equations to Differential Equations

In earlier classes, you solved equations like $x^2 + 1 = 0$ or $\sin^2 x - \cos x = 0$. The solution to such an equation is a number (real or complex) that, when substituted for the unknown $x$, makes the left-hand side equal to the right-hand side.

Now consider a differential equation, for example:

The crucial difference is that the unknown here is not a number but a function $y = \phi(x)$. A solution of this differential equation is a function $\phi$ such that when $\phi(x)$ and its derivatives are substituted into the equation, the left-hand side equals the right-hand side for all $x$ in the domain of interest.

The graph of such a function $y = \phi(x)$ is called a solution curve or integral curve of the differential equation.

---

The General Solution: A Family of Curves

Consider the function:

Let us verify that this is a solution of $\frac{d^2 y}{dx^2} + y = 0$.

›Proof

Verification of $y = a \sin(x + b)$ as a solution

First derivative:

$\frac{dy}{dx} = a \cos(x + b)$

Second derivative:

$\frac{d^2 y}{dx^2} = -a \sin(x + b)$

Substitute into the differential equation:

$\frac{d^2 y}{dx^2} + y = -a \sin(x + b) + a \sin(x + b) = 0$

Hence, the left-hand side equals the right-hand side for all $x$. Therefore, $y = a \sin(x + b)$ is indeed a solution.

Notice that this solution contains two arbitrary constants (parameters): $a$ and $b$. Because these constants can take any real value, the expression $y = a \sin(x + b)$ actually represents an entire family of curves — infinitely many solutions, one for each choice of $a$ and $b$.

This is called the general solution (or primitive) of the differential equation.

---

The Particular Solution: One Specific Curve

Now, suppose we assign specific numerical values to the arbitrary constants in the general solution. For example, let $a = 2$ and $b = \frac{\pi}{4}$. Then we obtain:

This function $\phi_1$ contains no arbitrary constants — only the particular values of the parameters. When we substitute $\phi_1$ and its derivatives into equation (3), the left-hand side still equals the right-hand side. So $\phi_1$ is also a solution.

---

Verifying a Solution