2. First verify whether y(x) satisfy the given Differential Equation. Then determine C dy a) x- dx +3y - 2x';y(x) = -x' +Cx*,y(2) = 1 b) y'+y tanx cosx, y(x) = (x+c)cosx, y(x) = 0 c) y' = 2y, y(x) = ce* -1, y(0) = 5
2. First verify whether y(x) satisfy the given Differential Equation. Then determine C dy a) x- dx +3y - 2x';y(x) = -x' +Cx*,y(2) = 1 b) y'+y tanx cosx, y(x) = (x+c)cosx, y(x) = 0 c) y' = 2y, y(x) = ce* -1, y(0) = 5
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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![## Differential Equation Verification and Determination of Constants
### Problem Statement:
**2. First verify whether \( y(x) \) satisfy the given Differential Equation. Then determine \( C \)**
#### a)
\[ x \frac{dy}{dx} + 3y = 2x^5; \, y(x) = -\frac{1}{4}x^5 + Cx^{-3}, y(2) = 1 \]
#### b)
\[ y' + y \tan x = \cos x, \, y(x) = (x + c) \cos x, y(\pi) = 0 \]
#### c)
\[ y' = 2y, \, y(x) = ce^x - 1, y(0) = 5 \]
### Details:
This problem involves verifying whether the provided functions \( y(x) \) satisfy their respective differential equations and then determining the constant \( C \) associated with them.
For each part (a, b, and c), we need to:
1. Substitute \( y(x) \) and its derivatives back into the given differential equation.
2. Confirm that the differential equation is satisfied.
3. Use the given initial or boundary condition to solve for the constant \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3861071f-90cb-4097-84b9-a523b25fee85%2Fceadba54-eb94-4dd2-b0f9-50e65436907d%2Fyevl6kq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Differential Equation Verification and Determination of Constants
### Problem Statement:
**2. First verify whether \( y(x) \) satisfy the given Differential Equation. Then determine \( C \)**
#### a)
\[ x \frac{dy}{dx} + 3y = 2x^5; \, y(x) = -\frac{1}{4}x^5 + Cx^{-3}, y(2) = 1 \]
#### b)
\[ y' + y \tan x = \cos x, \, y(x) = (x + c) \cos x, y(\pi) = 0 \]
#### c)
\[ y' = 2y, \, y(x) = ce^x - 1, y(0) = 5 \]
### Details:
This problem involves verifying whether the provided functions \( y(x) \) satisfy their respective differential equations and then determining the constant \( C \) associated with them.
For each part (a, b, and c), we need to:
1. Substitute \( y(x) \) and its derivatives back into the given differential equation.
2. Confirm that the differential equation is satisfied.
3. Use the given initial or boundary condition to solve for the constant \( C \).
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