Find the general solution of the given higher-order differential equation. dªy d²y dx² dx4 y(x) = - 8y = 0
Find the general solution of the given higher-order differential equation. dªy d²y dx² dx4 y(x) = - 8y = 0
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.1: Solutions Of Elementary And Separable Differential Equations
Problem 17E: Find the general solution for each differential equation. Verify that each solution satisfies the...
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![**Problem Statement:**
Find the general solution of the given higher-order differential equation.
\[ \frac{d^4 y}{dx^4} - 2 \frac{d^2 y}{dx^2} - 8y = 0 \]
\[ y(x) = \boxed{\phantom{answer}} \]
**Explanation:**
The given differential equation is a fourth-order homogeneous linear differential equation. The general solution involves finding the characteristic equation, solving for the roots, and constructing the general solution based on the nature of the roots (real or complex).
**Steps to Solve:**
1. **Find the characteristic equation:**
Replace \( \frac{d^4 y}{dx^4} \) with \( r^4 \), \( \frac{d^2 y}{dx^2} \) with \( r^2 \), and \( y \) with 1 to obtain the characteristic equation:
\[ r^4 - 2r^2 - 8 = 0 \]
2. **Solve the characteristic equation for \( r \):**
This is a quadratic equation in terms of \( r^2 \). Let \( z = r^2 \). The equation becomes:
\[ z^2 - 2z - 8 = 0 \]
Solving this quadratic equation using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives:
\[ z = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2} \]
Therefore,
\[ z = 4 \quad \text{or} \quad z = -2 \]
3. **Find the roots \( r \) from \( z \):**
For \( z = 4 \):
\[ r^2 = 4 \implies r = \pm 2 \]
For \( z = -2 \):
\[ r^2 = -2 \implies r = \pm i\sqrt{2} \] (complex roots)
4. **Construct the general solution:**
The general solution is constructed based on the roots of the characteristic equation](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4d7cfc5-c108-459e-b939-f2d2e6c37b5b%2Fef9af6b9-acd7-4c9f-b864-a27e3cd2de75%2F4ywboss_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the general solution of the given higher-order differential equation.
\[ \frac{d^4 y}{dx^4} - 2 \frac{d^2 y}{dx^2} - 8y = 0 \]
\[ y(x) = \boxed{\phantom{answer}} \]
**Explanation:**
The given differential equation is a fourth-order homogeneous linear differential equation. The general solution involves finding the characteristic equation, solving for the roots, and constructing the general solution based on the nature of the roots (real or complex).
**Steps to Solve:**
1. **Find the characteristic equation:**
Replace \( \frac{d^4 y}{dx^4} \) with \( r^4 \), \( \frac{d^2 y}{dx^2} \) with \( r^2 \), and \( y \) with 1 to obtain the characteristic equation:
\[ r^4 - 2r^2 - 8 = 0 \]
2. **Solve the characteristic equation for \( r \):**
This is a quadratic equation in terms of \( r^2 \). Let \( z = r^2 \). The equation becomes:
\[ z^2 - 2z - 8 = 0 \]
Solving this quadratic equation using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) gives:
\[ z = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2} \]
Therefore,
\[ z = 4 \quad \text{or} \quad z = -2 \]
3. **Find the roots \( r \) from \( z \):**
For \( z = 4 \):
\[ r^2 = 4 \implies r = \pm 2 \]
For \( z = -2 \):
\[ r^2 = -2 \implies r = \pm i\sqrt{2} \] (complex roots)
4. **Construct the general solution:**
The general solution is constructed based on the roots of the characteristic equation
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