A second-order differential equation is an equation that involves the second derivative of a function. In VCE Specialist Mathematics, the focus is on linear second-order differential equations with constant coefficients. These take the general form:
$$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)$$
where $a, b,$ and $c$ are real constants.
An equation is homogeneous if $f(x) = 0$. The equation becomes:
$$a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$$
An equation is non-homogeneous if $f(x) \neq 0$. To solve these, we find the general solution by combining the solution to the homogeneous part (the complementary function) and a specific solution (the particular integral).
KEY TAKEAWAY: The general solution to a second-order ODE always contains two arbitrary constants (usually $c_1$ and $c_2$ or $A$ and $B$). To find a particular solution, you must be provided with two pieces of information, such as $y(x_0) = y_0$ and $y’(x_0) = y’_0$.
If the differential equation is of the form $\frac{d^2y}{dx^2} = f(x)$, it can be solved by integrating twice with respect to $x$.
Steps:
1. Integrate once to find $\frac{dy}{dx} = \int f(x) \, dx + c_1$.
2. Integrate a second time to find $y = \int \left( \int f(x) \, dx + c_1 \right) dx + c_2$.
Example:
Solve $\frac{d^2y}{dx^2} = 12x^2$.
1. $\frac{dy}{dx} = \int 12x^2 \, dx = 4x^3 + c_1$
2. $y = \int (4x^3 + c_1) \, dx = x^4 + c_1x + c_2$
COMMON MISTAKE: Forgetting the first constant of integration ($c_1$) before the second integration. This constant becomes $c_1x$ in the final solution.
To solve $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$, we use the characteristic equation (also known as the auxiliary equation). We assume a solution of the form $y = e^{mx}$.
Substituting $y = e^{mx}$ into the ODE gives the quadratic:
$$am^2 + bm + c = 0$$
The nature of the roots ($m_1, m_2$) of this quadratic determines the form of the general solution:
| Nature of Roots | Roots | General Solution ($y_h$) |
|---|---|---|
| Distinct Real Roots | $m_1, m_2 \in \mathbb{R}$ | $y = Ae^{m_1x} + Be^{m_2x}$ |
| Repeated Real Roots | $m_1 = m_2 = m$ | $y = (A + Bx)e^{mx}$ |
| Complex Roots | $m = \alpha \pm \beta i$ | $y = e^{\alpha x}(A\cos(\beta x) + B\sin(\beta x))$ |
VCAA FOCUS: Complex roots frequently appear in the context of simple harmonic motion or damped oscillations. Ensure you are comfortable converting from the Cartesian form of the roots to the trigonometric general solution.
The general solution is given by:
$$y = y_h + y_p$$
Where:
* $y_h$ is the complementary function (solution to the homogeneous equation).
* $y_p$ is the particular integral (a specific solution that satisfies the non-homogeneous part).
We “guess” the form of $y_p$ based on $f(x)$ and then solve for the coefficients by substituting $y_p$ back into the original ODE.
| Form of $f(x)$ | Trial Particular Integral ($y_p$) |
|---|---|
| Polynomial of degree $n$ | $ax^n + bx^{n-1} + \dots + k$ |
| Exponential $ke^{nx}$ | $Ae^{nx}$ |
| Sine/Cosine $k\sin(nx)$ or $k\cos(nx)$ | $A\cos(nx) + B\sin(nx)$ |
Note: If the trial $y_p$ is already contained within the complementary function ($y_h$), multiply the trial solution by $x$ (or $x^2$) until it is linearly independent.
EXAM TIP: When finding $y_p$ for $f(x) = \sin(nx)$, you must include both $\sin(nx)$ and $\cos(nx)$ in your trial solution, even if only one appears in $f(x)$.
VCAA often asks students to verify that a given expression is a solution to a specific differential equation.
Process for Verification:
1. Calculate the first derivative ($\frac{dy}{dx}$) of the given expression.
2. Calculate the second derivative ($\frac{d^2y}{dx^2}$).
3. Substitute $y$, $\frac{dy}{dx}$, and $\frac{d^2y}{dx^2}$ into the Left Hand Side (LHS) of the ODE.
4. Simplify to show that the LHS equals the Right Hand Side (RHS).
STUDY HINT: Verification questions are “show that” questions. Ensure every step of your differentiation and substitution is clearly documented to earn full marks.
| Step | Action |
|---|---|
| 1 | Identify the equation as homogeneous or non-homogeneous. |
| 2 | Solve the characteristic equation $am^2 + bm + c = 0$ to find $y_h$. |
| 3 | If non-homogeneous, choose a trial $y_p$ based on $f(x)$. |
| 4 | Differentiate $y_p$ and substitute into the ODE to find specific constants. |
| 5 | Write the general solution: $y = y_h + y_p$. |
| 6 | Use initial conditions (e.g., $y(0)=k, y’(0)=j$) to find the values of $A$ and $B$. |
APPLICATION: Second-order ODEs are fundamental in physics for modeling Simple Harmonic Motion ($a = -n^2x$) and Damped Oscillations (e.g., a mass on a spring with air resistance), where the displacement $x$ is a function of time $t$.
