Consider the initial value problem y"+ y' – 12y = 0, y(0) = a, y'(0) = 5 Find the value of a so that the solution to the initial value problem approaches zero as t → 0 a =
Consider the initial value problem y"+ y' – 12y = 0, y(0) = a, y'(0) = 5 Find the value of a so that the solution to the initial value problem approaches zero as t → 0 a =
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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![**Consider the initial value problem:**
\[ y'' + y' - 12y = 0, \quad y(0) = \alpha, \quad y'(0) = 5 \]
**Find the value of \( \alpha \) so that the solution to the initial value problem approaches zero as \( t \to \infty \).**
\[ \alpha = \large \boxed{\phantom{000}} \]
**Explanation:**
In this problem, we are given a second-order linear differential equation with initial conditions. The goal is to determine the value of \( \alpha \), the initial value of \( y(t) \) at \( t = 0 \), such that the solution to the given differential equation approaches zero as \( t \) tends to infinity. This typically involves analyzing the characteristic equation of the differential equation, solving for the roots, and then using the initial conditions to find the specific solution that meets the criteria.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F62b092f3-63db-4d93-982f-67d0473d8e68%2F883b8064-daa5-4db9-8704-be1148a18c25%2Fue0l6t_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Consider the initial value problem:**
\[ y'' + y' - 12y = 0, \quad y(0) = \alpha, \quad y'(0) = 5 \]
**Find the value of \( \alpha \) so that the solution to the initial value problem approaches zero as \( t \to \infty \).**
\[ \alpha = \large \boxed{\phantom{000}} \]
**Explanation:**
In this problem, we are given a second-order linear differential equation with initial conditions. The goal is to determine the value of \( \alpha \), the initial value of \( y(t) \) at \( t = 0 \), such that the solution to the given differential equation approaches zero as \( t \) tends to infinity. This typically involves analyzing the characteristic equation of the differential equation, solving for the roots, and then using the initial conditions to find the specific solution that meets the criteria.
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