The image provides a differential equation problem with initial conditions. The problem is a third-order linear homogeneous differential equation with the following form:\[ y''' - 4y'' + 13y' + 50y = 0 \]Initial conditions are given as:- \( y(0) = 4 \)- \( y'(0) = 10 \)- \( y''(0) = 42 \)The task is to solve for \( y(t) \), and there is a box meant for the expression of \( y(t) \). **Problem Statement:**Solve the differential equation:\[ y^{(4)} + 18y'' + 81y = 0 \]with the initial conditions:\[ y(0) = -4, \quad y'(0) = 8, \quad y''(0) = 54, \quad y'''(0) = 0 \]**Explanation:**This is a fourth-order linear homogeneous differential equation with constant coefficients. The task is to find a function \( y(t) \) that satisfies this equation, given the specified initial conditions. The solution requires finding the characteristic equation, solving for the roots, and then constructing the general solution. The initial conditions are used to find specific values for the constants in the general solution.There are no graphs or diagrams included in this problem.

Given: The differential equation is y'''-4y''+13y'+50y=0 with initial conditions y0=4, y'0=10,…

Answered: y''' — 4y'' + 13y' + 50y 0 y(0) = 4,…

Math

Advanced Math

y''' — 4y'' + 13y' + 50y 0 y(0) = 4, y'(0) = 10, y’’(0) = 42 y(t) =

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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Question

The image provides a differential equation problem with initial conditions.

The problem is a third-order linear homogeneous differential equation with the following form:

\[ y''' - 4y'' + 13y' + 50y = 0 \]

Initial conditions are given as:
- \( y(0) = 4 \)
- \( y'(0) = 10 \)
- \( y''(0) = 42 \)

The task is to solve for \( y(t) \), and there is a box meant for the expression of \( y(t) \).

**Problem Statement:**

Solve the differential equation:

\[ y^{(4)} + 18y'' + 81y = 0 \]

with the initial conditions:

\[ y(0) = -4, \quad y'(0) = 8, \quad y''(0) = 54, \quad y'''(0) = 0 \]

**Explanation:**

This is a fourth-order linear homogeneous differential equation with constant coefficients. The task is to find a function \( y(t) \) that satisfies this equation, given the specified initial conditions.

The solution requires finding the characteristic equation, solving for the roots, and then constructing the general solution. The initial conditions are used to find specific values for the constants in the general solution.

There are no graphs or diagrams included in this problem.

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y (4) + 18y'' +81y = 0
4, y'(0)
y(0) :
=
=
8, y''(0) = 54, y'''(0) = 0

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