1. Find the general solution of dy dx +y=x
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![### Solving First Order Linear Differential Equations
#### Problem 1:
**Objective:** Find the general solution of the following differential equation:
\[ \frac{dy}{dx} + y = x \]
#### Solution:
To solve the first order linear differential equation \(\frac{dy}{dx} + y = x\), we need to follow these steps:
1. **Identify the Integrating Factor**:
The general form of a first-order linear differential equation is:
\[ \frac{dy}{dx} + p(x) y = q(x) \]
Here, \( p(x) = 1 \) and \( q(x) = x \).
2. **Calculate the Integrating Factor** (\( \mu(x) \)):
The integrating factor is given by:
\[ \mu(x) = e^{\int p(x) \, dx} \]
For \( p(x) = 1 \):
\[ \mu(x) = e^{\int 1 \, dx} = e^x \]
3. **Multiply the Differential Equation by the Integrating Factor**:
\[ e^x \frac{dy}{dx} + e^x y = x e^x \]
4. **Rewrite the Left-Hand Side** as a Single Derivative:
Notice that the left-hand side of the equation is the derivative of \( y e^x \):
\[ \frac{d}{dx} (y e^x) = x e^x \]
5. **Integrate Both Sides** with respect to \( x \):
\[ y e^x = \int x e^x \, dx \]
To solve the integral on the right-hand side, we use integration by parts:
\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x \]
6. **Combine and Isolate \( y \)**:
\[ y e^x = x e^x - e^x + C \]
\[ y = x - 1 + Ce^{-x} \]
Therefore, the general solution to the differential equation is:
\[ y = x - 1 + Ce^{-x} \]
where \( C \) is the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe02838e7-dc45-4607-88b3-6692c936110a%2F77e5e6d3-84ae-425b-9ecc-b31bab60e9b9%2Fmwsnd8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving First Order Linear Differential Equations
#### Problem 1:
**Objective:** Find the general solution of the following differential equation:
\[ \frac{dy}{dx} + y = x \]
#### Solution:
To solve the first order linear differential equation \(\frac{dy}{dx} + y = x\), we need to follow these steps:
1. **Identify the Integrating Factor**:
The general form of a first-order linear differential equation is:
\[ \frac{dy}{dx} + p(x) y = q(x) \]
Here, \( p(x) = 1 \) and \( q(x) = x \).
2. **Calculate the Integrating Factor** (\( \mu(x) \)):
The integrating factor is given by:
\[ \mu(x) = e^{\int p(x) \, dx} \]
For \( p(x) = 1 \):
\[ \mu(x) = e^{\int 1 \, dx} = e^x \]
3. **Multiply the Differential Equation by the Integrating Factor**:
\[ e^x \frac{dy}{dx} + e^x y = x e^x \]
4. **Rewrite the Left-Hand Side** as a Single Derivative:
Notice that the left-hand side of the equation is the derivative of \( y e^x \):
\[ \frac{d}{dx} (y e^x) = x e^x \]
5. **Integrate Both Sides** with respect to \( x \):
\[ y e^x = \int x e^x \, dx \]
To solve the integral on the right-hand side, we use integration by parts:
\[ \int x e^x \, dx = x e^x - \int e^x \, dx = x e^x - e^x \]
6. **Combine and Isolate \( y \)**:
\[ y e^x = x e^x - e^x + C \]
\[ y = x - 1 + Ce^{-x} \]
Therefore, the general solution to the differential equation is:
\[ y = x - 1 + Ce^{-x} \]
where \( C \) is the constant of integration.
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