Section 1.5 Order
In this section, we’ll explore a key concept in differential equations known as order. Think of the order of a differential equation as the number of layers or steps you need to peel away to reveal the original function. The more layers there are, the higher the order, and usually, the more complex the equation.
Example 9.
\(\ \ \)Give the order of the following differential equations:| \(\ds \frac{dy}{dx} + y = x\) |
\(\hspace{24em}\) Solution.
Order = 1. This is a first-order differential equation because the highest derivative is \(\frac{dy}{dx}\text{.}\)
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| \(\ds x^2 y'' + y''' = \sin(x) + y^8\) |
\(\hspace{24em}\) Solution.
Order = 3. This is a third-order differential equation because the highest derivative is \(y'''\text{.}\)
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| \(\ds \frac{d^2 A}{dt^2} + \frac{dA}{dt} + A = 17\) |
\(\hspace{24em}\) Solution.
Order = 2 This is a second-order differential equation because the highest derivative is \(\frac{d^2 A}{dt^2}\text{.}\)
|
Note 10. Caution, Don’t Confuse Exponents with Derivatives.
It’s important to distinguish between exponents and derivatives. For example, in the second example, \(y^8\) indicates that the dependent variable is raised to the 8th power, but this is not related to the order of the differential equation.
Reading Questions Check your Understanding
1. Which of the following equations is a third-order differential equation?
- \(\frac{d^3y}{dx^3} + x^2y = 0\)
- Correct! The highest derivative here is the third derivative, making it a third-order differential equation.
- \(\frac{d^2y}{dx^2} + y' = \sin x\)
- Incorrect. This is a second-order differential equation.
- \(y'' + y' + y = 0\)
- Incorrect. This is a second-order differential equation.
- \(y' + y = x\)
- Incorrect. This is a first-order differential equation.
Which of the following equations is a third-order differential equation?
2. The order of a differential equation is determined by the number of terms it contains.
- True
- Incorrect. The order is based on the highest derivative, regardless of the number of terms.
- False
- Correct! The order is determined by the highest derivative, not the number of terms.
The order of a differential equation is determined by the number of terms it contains
3. The differential equation \(\ds (1 - x)y'' - x^4y' + y^5 = \cos x \) has order .
4. The differential equation \(\ds y''' + 3y' - 4y = 0 \) has order .
5. The differential equation \(\ds t^3 \frac{d^2y}{dt^2} + \frac{d^5y}{dt^5} = \sin(t) \) has order .
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