DIFFERENTIAL EQUATIONS 07
Second-Order Differential Equations
A second-order differential equation contains a second derivative and often describes acceleration, curvature or oscillation.
The main idea
A second-order ordinary differential equation contains
as its highest derivative.
A general second-order ODE may be written as
\[ F\left( x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2} \right)=0. \]1. What does the second derivative mean?
The first derivative
\[ \frac{dy}{dx} \]describes how \(y\) changes.
The second derivative
\[ \frac{d^2y}{dx^2} \]describes how that first derivative changes.
2. Physical interpretation
If
\[ s(t) \]is position, then
\[ \frac{ds}{dt} \]is velocity and
\[ \frac{d^2s}{dt^2} \]is acceleration.
3. A simple second-order equation
Consider
\[ \frac{d^2y}{dx^2}=6x. \]Integrate once:
\[ \frac{dy}{dx} = 3x^2+C_1. \]Integrate again:
\[ y = x^3+C_1x+C_2. \]Therefore,
4. Why two constants appear
Each integration introduces an arbitrary constant.
A second-order equation therefore usually produces two constants:
\[ C_1 \qquad\text{and}\qquad C_2. \]This is why two independent conditions are generally needed to determine a particular solution.
5. Initial conditions
For a second-order equation, we may be given
\[ y(0)=a \]and
\[ y'(0)=b. \]The first condition gives the initial value of the function. The second gives its initial rate of change.
6. Example with initial conditions
Solve
\[ y''=6x, \]with
\[ y(0)=2, \qquad y'(0)=1. \]From the general solution,
\[ y'=3x^2+C_1. \]Using
\[ y'(0)=1, \]we obtain
\[ C_1=1. \]So
\[ y=x^3+x+C_2. \]Using
\[ y(0)=2, \]we get
\[ C_2=2. \]Therefore,
7. Linear second-order equations
A common form is
where \(a\), \(b\), and \(c\) may be constants.
If
\[ g(x)=0, \]the equation is called homogeneous.
8. Homogeneous constant-coefficient equation
Consider
\[ y''-3y'+2y=0. \]We try a solution of the form
\[ y=e^{rx}. \]Then
\[ y'=re^{rx} \]and
\[ y''=r^2e^{rx}. \]Substitute:
\[ r^2e^{rx} - 3re^{rx} + 2e^{rx} = 0. \]Since
\[ e^{rx}\neq0, \]we obtain
9. The characteristic equation
The polynomial equation
\[ r^2-3r+2=0 \]is called the characteristic equation.
Factorising,
\[ (r-1)(r-2)=0. \]So the roots are
\[ r=1, \qquad r=2. \]10. Distinct real roots
When the characteristic equation has two distinct real roots
\[ r_1, \qquad r_2, \]the general solution is
For the previous example,
\[ \boxed{ y=C_1e^x+C_2e^{2x} } \]11. Why the two solutions can be combined
The equation is linear and homogeneous.
If \(y_1\) and \(y_2\) are solutions, then
\[ C_1y_1+C_2y_2 \]is also a solution.
This is called the principle of superposition.
12. Repeated root
Suppose the characteristic equation has a repeated root \(r\).
For example,
\[ y''-2y'+y=0. \]The characteristic equation is
\[ r^2-2r+1=0. \]So
\[ (r-1)^2=0. \]There is one repeated root:
\[ r=1. \]The general solution is
13. Complex roots
Consider
\[ y''+y=0. \]The characteristic equation is
\[ r^2+1=0. \]So
\[ r=\pm i. \]The real-valued general solution is
14. Why sine and cosine appear
For
\[ y''+y=0, \]we can write
\[ y''=-y. \]So the second derivative is always the negative of the function itself.
Sine and cosine have exactly this property:
\[ \frac{d^2}{dx^2}\sin x=-\sin x, \] \[ \frac{d^2}{dx^2}\cos x=-\cos x. \]That is why oscillations naturally involve trigonometric functions.
15. Simple harmonic motion
A common model is
where \(\omega>0\).
The solution is
\[ x(t) = A\cos(\omega t) + B\sin(\omega t). \]This describes repeated oscillation.
16. Visual idea
17. Non-homogeneous equations
If
\[ ay''+by'+cy=g(x) \]with
\[ g(x)\neq0, \]the equation is non-homogeneous.
Its solution is usually written as
where \(y_h\) solves the associated homogeneous equation and \(y_p\) is one particular solution of the full equation.
18. Example of forcing
An equation such as
\[ y''+y=\cos t \]contains an external forcing term
\[ \cos t. \]The system has its own natural behaviour from
\[ y''+y=0, \]but it is also driven by the external input.
19. Why second-order equations matter
Second-order equations appear whenever acceleration or curvature is important.
Examples include:
\[ \text{mechanical motion}, \] \[ \text{springs}, \] \[ \text{oscillations}, \] \[ \text{electrical circuits}, \]and many other dynamical systems.
20. The practical structure
For linear constant-coefficient equations, the characteristic equation provides a systematic way to find the general solution.