Find the general solution of the following differential equation. Primes denote derivatives with respect to x.
left parenthesis x plus 8 y right parenthesis y prime equals 5 x minus y


To find the general solution of the differential equation:

\[
(x + 8y) y' = 5x - y
\]

where \(y' = \frac{dy}{dx}\), let's rewrite the equation in a more standard form. Start by rewriting it as:

\[
y' = \frac{5x - y}{x + 8y}
\]

This differential equation is not immediately recognizable as a standard type (like separable, linear, or exact). However, it suggests the possibility of a substitution to simplify it. A good substitution to try here is:

\[
v = x + 8y \implies y = \frac{v - x}{8}
\]

Differentiating \(y\) with respect to \(x\):

\[
y' = \frac{dv}{dx} \cdot \frac{1}{8} - \frac{1}{8}
\]

Substitute these expressions into the original differential equation:

\[
v \cdot \left(\frac{1}{8} \frac{dv}{dx} - \frac{1}{8}\right) = 5x - \frac{v - x}{8}
\]

Simplify this equation:

\[
\frac{v}{8} \frac{dv}{dx} - \frac{v}{8} = 5x - \frac{v}{8} + \frac{x}{8}
\]

Multiply the entire equation by 8 to clear the denominators:

\[
v \frac{dv}{dx} - v = 40x - v + x
\]

Combine like terms:

\[
v \frac{dv}{dx} = 41x
\]

Separate the variables:

\[
v \, dv = 41x \, dx
\]

Integrate both sides:

\[
\int v \, dv = \int 41x \, dx
\]

\[
\frac{v^2}{2} = \frac{41x^2}{2} + C
\]

Solve for \(v^2\):

\[
v^2 = 41x^2 + 2C
\]

Substitute back \(v = x + 8y\):

\[
(x + 8y)^2 = 41x^2 + 2C
\]

Take the square root of both sides:

\[
x + 8y = \pm \sqrt{41x^2 + 2C}
\]

Solve for \(y\):

\[
8y = -x \pm \sqrt{41x^2 + 2C}
\]

\[
y = \frac{-x \pm \sqrt{41x^2 + 2C}}{8}
\]

This is the general solution of the given differential equation.

### Would you like more details or have any questions?

Here are some related questions for further exploration:

1. What are the standard forms of differential equations and their methods of solution?
2. How do substitutions help simplify certain differential equations?
3. What is the significance of constants of integration in differential equations?
4. How can initial conditions be used to find a particular solution from a general solution?
5. What are other common methods for solving first-order differential equations?

**Tip:** When solving differential equations, always check if there is a substitution that can simplify the expression, especially when the equation involves combinations of variables.

Find the general solution of the following differential equation. Primes denote derivatives with respect to x: (x + 8y) y' = 5x - y

Explore the general solution of the differential equation (x + 8y) y' = 5x - y using substitution and integration techniques. This solution covers key concepts in differential equations, including variable substitution and integration methods. Ideal for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation