238. [T] sin(x)y + cos(x) = 2x 239. [T] √x² + 1y = y +2 240. [T] xy+2r2y=x+1 Solve the following initial-value problems by using integrating factors. 241. y' + y = x, y(0) = 3 242. y' = y + 2x², y(0) = 0 243. xy' = y - 3x³, y(1) = 0 244. x²y = xy-lnx, y(1) = 1 245. (1+x²)y=y-1, y(0) = 0 246. xy = y + 2xlnx, y(1) = 5 wing B
238. [T] sin(x)y + cos(x) = 2x 239. [T] √x² + 1y = y +2 240. [T] xy+2r2y=x+1 Solve the following initial-value problems by using integrating factors. 241. y' + y = x, y(0) = 3 242. y' = y + 2x², y(0) = 0 243. xy' = y - 3x³, y(1) = 0 244. x²y = xy-lnx, y(1) = 1 245. (1+x²)y=y-1, y(0) = 0 246. xy = y + 2xlnx, y(1) = 5 wing B
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Solve 241 ,243
![**Educational Website Content**
### Topics in Differential Equations
**Problem Set: Initial-Value Problems with Integrating Factors**
This section contains a set of differential equations to be solved using the method of integrating factors. These problems are essential for understanding the application of integrating factors in solving linear first-order differential equations with initial conditions.
**Problem List:**
1. **Problem 241:**
\[
y' + y = x, \quad y(0) = 3
\]
2. **Problem 242:**
\[
y' = y + 2x^2, \quad y(0) = 0
\]
3. **Problem 243:**
\[
xy' = y - 3x^3, \quad y(1) = 0
\]
4. **Problem 244:**
\[
x^2 y' = xy - \ln x, \quad y(1) = 1
\]
5. **Problem 245:**
\[
(1 + x^2)y' = y - 1, \quad y(0) = 0
\]
6. **Problem 246:**
\[
xy' = y + 2x \ln x, \quad y(1) = 5
\]
**Instructions:**
- Solve each differential equation using the integrating factor method.
- Ensure that the solution satisfies the given initial condition.
**Note:** This problem set is designed to develop skills in solving initial-value problems using a systematic approach. The technique of integrating factors is a powerful tool for finding analytic solutions to linear first-order differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03fd5f25-eded-4046-9c65-9e89b5e28b89%2Fe892fed2-da4a-4c2a-87c1-bc9d7b410ff7%2Fyozt0c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Website Content**
### Topics in Differential Equations
**Problem Set: Initial-Value Problems with Integrating Factors**
This section contains a set of differential equations to be solved using the method of integrating factors. These problems are essential for understanding the application of integrating factors in solving linear first-order differential equations with initial conditions.
**Problem List:**
1. **Problem 241:**
\[
y' + y = x, \quad y(0) = 3
\]
2. **Problem 242:**
\[
y' = y + 2x^2, \quad y(0) = 0
\]
3. **Problem 243:**
\[
xy' = y - 3x^3, \quad y(1) = 0
\]
4. **Problem 244:**
\[
x^2 y' = xy - \ln x, \quad y(1) = 1
\]
5. **Problem 245:**
\[
(1 + x^2)y' = y - 1, \quad y(0) = 0
\]
6. **Problem 246:**
\[
xy' = y + 2x \ln x, \quad y(1) = 5
\]
**Instructions:**
- Solve each differential equation using the integrating factor method.
- Ensure that the solution satisfies the given initial condition.
**Note:** This problem set is designed to develop skills in solving initial-value problems using a systematic approach. The technique of integrating factors is a powerful tool for finding analytic solutions to linear first-order differential equations.
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