y' + x(y² + y) = 0, y(2) = 1
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 Equations: Separable and Linear
**Solve the following: Each question is either separable or linear:**
1. \(\frac{dy}{dx} + P(x)y = Q(x)\)
2. \(\frac{dy}{dx} = g(x)h(y)\)
Let's delve into each type of differential equation for clearer understanding and step-by-step solutions.
---
Below are the steps for solving each category:
#### Separable Differential Equations:
A separable differential equation is one that can be expressed in the form:
\[ \frac{dy}{dx} = g(x)h(y) \]
To solve:
1. Separate the variables: \(\frac{1}{h(y)} dy = g(x) dx\).
2. Integrate both sides: \(\int \frac{1}{h(y)} dy = \int g(x) dx\).
3. Solve the resulting equations to find \( y \) in terms of \( x \).
#### Linear Differential Equations:
A linear differential equation has the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
To solve:
1. Find the integrating factor, \(\mu(x) = e^{\int P(x) dx}\).
2. Multiply through by the integrating factor: \( \mu(x) \frac{dy}{dx} + \mu(x) P(x)y = \mu(x) Q(x) \).
3. Recognize that the left side is the derivative of \( \mu(x)y \).
4. Integrate both sides: \(\int d(\mu(x)y) = \int \mu(x) Q(x) dx\).
5. Solve for \( y \) explicitly.
Feel free to click on each equation for detailed examples and step-by-step solutions!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7c2325-139a-41dc-93c7-07b552835718%2F0ed699dd-211e-486f-9d7d-e109f29a116d%2Fq97g7i_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Separable and Linear
**Solve the following: Each question is either separable or linear:**
1. \(\frac{dy}{dx} + P(x)y = Q(x)\)
2. \(\frac{dy}{dx} = g(x)h(y)\)
Let's delve into each type of differential equation for clearer understanding and step-by-step solutions.
---
Below are the steps for solving each category:
#### Separable Differential Equations:
A separable differential equation is one that can be expressed in the form:
\[ \frac{dy}{dx} = g(x)h(y) \]
To solve:
1. Separate the variables: \(\frac{1}{h(y)} dy = g(x) dx\).
2. Integrate both sides: \(\int \frac{1}{h(y)} dy = \int g(x) dx\).
3. Solve the resulting equations to find \( y \) in terms of \( x \).
#### Linear Differential Equations:
A linear differential equation has the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
To solve:
1. Find the integrating factor, \(\mu(x) = e^{\int P(x) dx}\).
2. Multiply through by the integrating factor: \( \mu(x) \frac{dy}{dx} + \mu(x) P(x)y = \mu(x) Q(x) \).
3. Recognize that the left side is the derivative of \( \mu(x)y \).
4. Integrate both sides: \(\int d(\mu(x)y) = \int \mu(x) Q(x) dx\).
5. Solve for \( y \) explicitly.
Feel free to click on each equation for detailed examples and step-by-step solutions!
![### Differential Equations Example
In this example, we examine the following first-order ordinary differential equation with an initial condition:
\[ y' + x(y^2 + y) = 0, \quad y(2) = 1 \]
This differential equation can be analyzed and solved to find the function \( y(x) \) that satisfies both the differential equation and the initial condition given at \( x = 2 \) where \( y = 1 \).
### Explanation
- **First-Order ODE**: This is a first-order ordinary differential equation, meaning it involves the first derivative of the function \( y \) with respect to \( x \), represented as \( y' \).
- **Initial Condition**: The initial condition \( y(2) = 1 \) specifies the value of the function \( y \) at \( x = 2 \).
Students are encouraged to solve the equation using appropriate methods such as separation of variables, integrating factors, or numerical methods to find the explicit form of \( y(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7c2325-139a-41dc-93c7-07b552835718%2F0ed699dd-211e-486f-9d7d-e109f29a116d%2F3yfesjd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations Example
In this example, we examine the following first-order ordinary differential equation with an initial condition:
\[ y' + x(y^2 + y) = 0, \quad y(2) = 1 \]
This differential equation can be analyzed and solved to find the function \( y(x) \) that satisfies both the differential equation and the initial condition given at \( x = 2 \) where \( y = 1 \).
### Explanation
- **First-Order ODE**: This is a first-order ordinary differential equation, meaning it involves the first derivative of the function \( y \) with respect to \( x \), represented as \( y' \).
- **Initial Condition**: The initial condition \( y(2) = 1 \) specifies the value of the function \( y \) at \( x = 2 \).
Students are encouraged to solve the equation using appropriate methods such as separation of variables, integrating factors, or numerical methods to find the explicit form of \( y(x) \).
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