https://file.aiquickdraw.com/mathbot/file/17294672187736a1ta43s.jpg

The given differential equation is:

\[
xy' = 13y + x^{14} \cos(x)
\]

To solve this, we will try to solve the equation by standard techniques such as solving it as a first-order linear differential equation.

### Step 1: Rewrite the equation
We divide the whole equation by \(x\) to simplify it:

\[
y' - \frac{13}{x}y = x^{13} \cos(x)
\]

This is a first-order linear differential equation of the form:

\[
y' + P(x)y = Q(x)
\]

where \(P(x) = -\frac{13}{x}\) and \(Q(x) = x^{13} \cos(x)\).

### Step 2: Find the integrating factor
The integrating factor \( \mu(x) \) is given by:

\[
\mu(x) = e^{\int P(x)\, dx} = e^{\int -\frac{13}{x}\, dx} = e^{-13 \ln|x|} = |x|^{-13}
\]

### Step 3: Multiply both sides of the equation by the integrating factor
Multiplying the entire differential equation by \( |x|^{-13} \):

\[
|x|^{-13}y' - \frac{13}{x}|x|^{-13}y = |x|^{-13} x^{13} \cos(x)
\]

This simplifies to:

\[
\frac{d}{dx}\left(|x|^{-13}y\right) = \cos(x)
\]

### Step 4: Integrate both sides
Now, integrate both sides with respect to \(x\):

\[
|x|^{-13}y = \int \cos(x)\, dx = \sin(x) + C
\]

where \(C\) is the constant of integration.

### Step 5: Solve for \(y\)
Finally, solve for \(y\):

\[
y = |x|^{13}(\sin(x) + C)
\]

Thus, the general solution to the differential equation is:

\[
y = x^{13}(\sin(x) + C)
\]

Would you like further clarification or more details on any step?

Here are some related questions:
1. How do you derive the integrating factor in linear differential equations?
2. What are other techniques for solving first-order linear differential equations?
3. Can this solution method be applied to nonlinear equations?
4. How does the presence of \(\cos(x)\) affect the behavior of the solution?
5. What is the significance of the constant \(C\) in the solution?

**Tip**: When solving first-order linear differential equations, always look for the integrating factor to simplify the equation into an exact differential.

Find the general solution of the differential equation: xy' = 13y + x^14 cos(x).

This page explains how to solve the first-order linear differential equation xy' = 13y + x^14 cos(x) using the integrating factor method. It involves key concepts in solving linear differential equations, such as finding the integrating factor and using trigonometric functions in integration.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation