Basics Of Second Order Differential Equation



A second order differential equation is of the form $a \frac{d^{2}y}{dx^{2}} + b \frac{dy}{dx} + b \frac{dy}{dx} + cy = f(x)$.

$f(x)$ is called a source term or forcing function.

The differential equation is called

  • a homogeneous equation IF $f(x) = 0$
  • non-homogeneous IF $f(x)$ is not 0.

The steps involved in solving a homogeneous equation and non-homogeneous are quite similar, with the non-homogeneous one requiring more work. (More about this later)

Steps to solving a homogeneous equation:

  1. Rewrite the given differential equation $a \frac{d^{2}y}{dx^{2}} + b \frac{dy}{dx} + b \frac{dy}{dx} + cy = f(x)$ as $(aD^2 + bD + c)y = 0$.
  2. Substitute m for D and solve the auxiliary equation $am^2 + bm + c = 0$
    1. If the roots of the auxiliary equation are real and different, e.g. $m = \alpha$ and $m = \beta$, then the general solution is: $$y = A e^{\alpha x} + B e^{\beta x}$$
    2. If the roots are real and equal, e.g. $m = \alpha$, twice, then the general solution is $$y = (Ax + B)e^{\alpha x}$$.
    3. If the roots are complex, e.g. $m = \alpha \pm \beta i$, then the general solution is $$y = e^{\alpha x} ( C \cos \beta x + D \sin \beta x)$$
  3. If the particular solution of a differential equation is required then substitute the given boundary conditions to find the unknown constants.

 

Non-homogeneous 

The general solution for a non-homogeneous second order differential equation is given by y = complementary function + particular integral, which is y = u + v.

By following the four steps above, the complementary function for the non-homogeneous differential equation is found.

Note: Treat f(x) as 0 when solving for the complementary function.

 

Finding the particular integral

There are no hard and fast rule in finding the particular integral. (It involves some guessing) I will show you a few examples below.

Example 1: Solve $\frac{d^{2}y}{dx^2} – 4 \frac{dy}{dx} + 4y = 4x + 3cos 2x$

$$(D^{2} – 4D + 4)y = 0$$

The auxiliary equation is $m^{2} – 4m + 4 = 0$
$$(m – 2)(m-2) = 0$$ $$m = 2$$

The complementary function is $u = (Ax + B)e^{2x}$.

Make a guess! Let the particular integral be $ v = ax + b + C \cos 2x + D \sin 2x$

Note: Normally, you would try the most general form of the source term. E.g. The source term contains x, hence you make a guess that the final solution must contain the most general form of x, i.e. ax + b. The most general form of cos x is sin x and cos x.

$$(D^{2} – 4D + 4)v = 4x + 3 \cos 2x$$ $$D(v) = a – 2C \sin 2x + 2D \cos 2x$$ $$D^{2}(v) = -4C \cos 2x – 4D \sin 2x$$

Substitute D(v) and $D^{2}(v)$ into $(D^{2} – 4D + 4)v = 4x + 3 \cos 2x$

$$4ax – 4a + 4b + (-4C – 8D + 4C)\cos 2x + (-4D + 8C + 4D)\sin 2x = 4x + 3\cos 2x$$

By comparing the coefficients,
a = 1, b = 1, C = 0, $D = -\frac{3}{8}$

Hence, the particular integral is $v = x + 1 – \frac{3}{8} \sin 2x$

The general solution is:
$$y = (Ax + B)e^{2x} + x + 1 – \frac{3}{8} \sin 2x$$

______________________________________________
The “guess” is called an Ansatz.

In physics and mathematics, an ansatz is an educated guess that is verified later by its results.

-From Wikipedia

 

This is the end of the basic walkthrough for Second Order Differential Equation.

Back To Second Order Differential Equation

Back To University Year 1 Physics Notes



Mini Physics

Administrator of Mini Physics. If you spot any errors or want to suggest improvements, please contact us. If you like the content in this site, please recommend this site to your friends!



Leave a Comment

This site uses Akismet to reduce spam. Learn how your comment data is processed.