Second Order Differential Equation

Second Order Differential Equation
Written by: Jack Methew

Jack Methew knows that successful students become successful adults. This is her 15th year at Edison Elementary School and her 10th year teaching fourth grade. So far, fourth grade is her favorite grade to teach! Mrs. Carroll was the 2011 Newell Unified School District Teacher of the Year, and received her National Board Certification in 2013. She loves science and majored in biology at Arizona State University, where she also earned her teaching credential and Master of Education degree. Mrs. Carroll is excited to begin the best year ever!

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We can solve the second order differential equation of the type

$$\frac{d^2y}{d^2x}+p(x)\frac{dy}{dx}+Q(x)y=f(x)$$

Fact: Where p(x),Q(x) and f(x) are functions of x , by using

Variation of Parameters

Which only works when f(x) is polynomial, exponential, sine, cosine or a linear combination of those.

Method of Undetermined Coefficient

Which is shortest but works on a wider range of functions.

Fact: But here we start our discussion by learning the case where (x)=0,(it is a homogeneous equation).

$$\frac{d^2y}{d^2x}+p(x)\frac{dy}{dx}+Q(x)y=f(x)$$

And also where a function P(x) and Q(x) are constants p and q

Let’s try to solve them:

e to be rescue

We are going to use a special property of derivative of exponential function

Fact: At any point the slope (derivative ) of ex equal of the value ex:

And when we introduce a value “r” like this:

$$f(x)\;=\;e^{rx}$$

we find

  • The first derivative is

    $$f'(x)\;=\;rx$$

  • The second derivative is

    $$f"(x)\;=\;r^2e^{rx}$$

Fact: The first and second derivative are both functions of f(x)

This is going to be helpful to us.

Example

$$\frac{d^2y}{dx^2}+\frac{dy}{dx}-6x=0$$

Let

$$y=e^{rx}$$

So, we have:

  • $$\frac{dy}{dx}=re^{rx}$$
  • $$\frac{d^2y}{dx^2}=r^2e^{rx}$$

Substitute these into the given equation:

$$r^2e^{rx}+re^{rx-6}e^{rx}=0$$

Simplify:

$$e^{rx}(r^2+r-6)=0$$

$$r^2+r-6=0$$

We have reduced the differential equation to an ordinary quadratic equation.

The quadratic equation is given the special name of the characteristic equation.

We can factor this equation:

$$r^2+r-6$$

$$(r-1)(r+3)=0$$

So,

$$r=2,3$$

And so, we have two solution:

$$y=e^2x$$

$$e^{-3x}$$

General Solution:

$$y=Ae^2x+Be^{-3x}$$