Find the general solution to the equation y(4) – 16y" + 64y" = 0. NOTE: Use С), С9, Сз, апd сд as arbitrarу constants. y =
Find the general solution to the equation y(4) – 16y" + 64y" = 0. NOTE: Use С), С9, Сз, апd сд as arbitrarу constants. y =
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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![### Find the General Solution to the Differential Equation
Given the following differential equation:
\[ y^{(4)} - 16y''' + 64y'' = 0 \]
You are required to find the general solution to this equation.
#### Hint:
**Use \( c_1, c_2, c_3, \) and \( c_4 \) as arbitrary constants.**
\[ y = \boxed{\phantom{y}} \]
This is a fourth-order linear homogeneous differential equation with constant coefficients. Solving such equations typically involves finding the characteristic equation and its roots, followed by constructing the general solution using those roots.
To solve for the general solution:
1. Formulate the characteristic equation associated with the given differential equation.
2. Solve for the roots of the characteristic equation.
3. Construct the general solution based on the nature of the roots found (real and distinct, repeated, complex conjugates).
Note that using the arbitrary constants \( c_1, c_2, c_3, \) and \( c_4 \) will help in expressing the general solution appropriately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5362cbea-9427-4b2b-99cb-57a208edd853%2F8b5d0d5d-b37d-450c-bf5d-551dda2b2be6%2F1i31i2k_processed.png&w=3840&q=75)
Transcribed Image Text:### Find the General Solution to the Differential Equation
Given the following differential equation:
\[ y^{(4)} - 16y''' + 64y'' = 0 \]
You are required to find the general solution to this equation.
#### Hint:
**Use \( c_1, c_2, c_3, \) and \( c_4 \) as arbitrary constants.**
\[ y = \boxed{\phantom{y}} \]
This is a fourth-order linear homogeneous differential equation with constant coefficients. Solving such equations typically involves finding the characteristic equation and its roots, followed by constructing the general solution using those roots.
To solve for the general solution:
1. Formulate the characteristic equation associated with the given differential equation.
2. Solve for the roots of the characteristic equation.
3. Construct the general solution based on the nature of the roots found (real and distinct, repeated, complex conjugates).
Note that using the arbitrary constants \( c_1, c_2, c_3, \) and \( c_4 \) will help in expressing the general solution appropriately.
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