e order of the given ordinary differential equation, and classify it as linear or non-linear. 2 (n) + a df – sin(x) = 0 dx4 (sin(x)-x³) e2x — x²y"(x) = e5x sin(2x) Зау' + Зу'' = 8b2 d³q tan(2) p²q-q sin(p) Note: The symbols a and b are non-zero constants. If a variable appears alone in parenthesis immediately after the symbol for the function (or one of its derivatives), then the parenthesis does not indicate multiplication. The parentheses indicate that the variable is the independent variable of the function.
e order of the given ordinary differential equation, and classify it as linear or non-linear. 2 (n) + a df – sin(x) = 0 dx4 (sin(x)-x³) e2x — x²y"(x) = e5x sin(2x) Зау' + Зу'' = 8b2 d³q tan(2) p²q-q sin(p) Note: The symbols a and b are non-zero constants. If a variable appears alone in parenthesis immediately after the symbol for the function (or one of its derivatives), then the parenthesis does not indicate multiplication. The parentheses indicate that the variable is the independent variable of the function.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Ordinary Differential Equations: Order and Classification
#### Objective:
State the order of the given ordinary differential equation and classify it as linear or non-linear.
**A.3.a.**
\[ \left( \frac{d^3 f}{dx^3} \right)^2 + a \frac{d^4 f}{dx^4} - \sin(\pi x) = 0 \]
**A.3.b.**
\[ \frac{(\sin(x) - x^3)}{e^{2x}} - x^2 y''(x) = e^{5x} \sin(2x) \]
**A.3.c.**
\[ 3a y' + 3y'' = 8b^2 \]
**A.3.d.**
\[ \frac{d^5 q}{dp^5} \tan(2q) = \frac{p^{2q - q} \sin(p)}{e^{ap}} \]
**A.3.e.**
\[ -9\sinh(at) = -z''' - az' t^{-2} \]
**A.3.f.**
\[ r'' = -a^5 x + r\sin(rx) \]
**A.3.g.**
\[ y' x^2 + y'' = 2 - x^2 \cos(y) \]
#### Note:
- The symbols \(a\) and \(b\) are non-zero constants.
- If a variable appears alone in parenthesis immediately after the symbol for the function (or one of its derivatives), then the parenthesis does not indicate multiplication. The parentheses indicate that the variable is the independent variable of the function.
---
In the above equations:
- **Order:** The highest derivative of the unknown function present in the equation determines the order of the differential equation.
- **Linearity:** A differential equation is linear if the unknown function and its derivatives appear to the power of one (i.e., they are not multiplied together). Non-linear differential equations involve powers or products of the function or its derivatives, or function arguments in non-linear functions.
Use this guide to further analyze and classify the given ordinary differential equations based on their order and whether they are linear or non-linear.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2e521a1e-f4ee-4c02-af71-8dfc2cc184d3%2F072567d6-77fc-4f15-b733-b6af7d54e974%2Fn0iobmd_processed.png&w=3840&q=75)
Transcribed Image Text:### Ordinary Differential Equations: Order and Classification
#### Objective:
State the order of the given ordinary differential equation and classify it as linear or non-linear.
**A.3.a.**
\[ \left( \frac{d^3 f}{dx^3} \right)^2 + a \frac{d^4 f}{dx^4} - \sin(\pi x) = 0 \]
**A.3.b.**
\[ \frac{(\sin(x) - x^3)}{e^{2x}} - x^2 y''(x) = e^{5x} \sin(2x) \]
**A.3.c.**
\[ 3a y' + 3y'' = 8b^2 \]
**A.3.d.**
\[ \frac{d^5 q}{dp^5} \tan(2q) = \frac{p^{2q - q} \sin(p)}{e^{ap}} \]
**A.3.e.**
\[ -9\sinh(at) = -z''' - az' t^{-2} \]
**A.3.f.**
\[ r'' = -a^5 x + r\sin(rx) \]
**A.3.g.**
\[ y' x^2 + y'' = 2 - x^2 \cos(y) \]
#### Note:
- The symbols \(a\) and \(b\) are non-zero constants.
- If a variable appears alone in parenthesis immediately after the symbol for the function (or one of its derivatives), then the parenthesis does not indicate multiplication. The parentheses indicate that the variable is the independent variable of the function.
---
In the above equations:
- **Order:** The highest derivative of the unknown function present in the equation determines the order of the differential equation.
- **Linearity:** A differential equation is linear if the unknown function and its derivatives appear to the power of one (i.e., they are not multiplied together). Non-linear differential equations involve powers or products of the function or its derivatives, or function arguments in non-linear functions.
Use this guide to further analyze and classify the given ordinary differential equations based on their order and whether they are linear or non-linear.
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