Use the method for solving homogeneous equations to solve the following differential equation. (12y² - xy) dx + x² dy=0 Ignoring lost solutions, if any, the general solution is y= (Type an expression using x as the variable.)
Use the method for solving homogeneous equations to solve the following differential equation. (12y² - xy) dx + x² dy=0 Ignoring lost solutions, if any, the general solution is y= (Type an expression using x as the variable.)
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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![**Solving Homogeneous Differential Equations**
To solve the given differential equation using the method for homogeneous equations, follow these steps:
### Differential Equation
\[ (12y^2 - xy) \, dx + x^2 \, dy = 0 \]
### Task
Ignoring lost solutions, if any, the general solution is \( y = \) [___].
(Type an expression using \( x \) as the variable.)
### Explanation of Steps
1. **Rewrite the Differential Equation**: Express the differential equation in the standard form for homogeneous equations, if not already done.
2. **Substitution to Simplify**: Usually, substitute \( v = \frac{y}{x} \) and hence \( y = vx \). Consequently, \( dy = v \, dx + x \, dv \). Replace \( y \) and \( dy \) in the original equation.
3. **Separation of Variables**: After substituting, the equation should be separable. Separate the variables and integrate both sides to find the relationship between \( x \) and \( y \).
4. **Solve for \( y \)**: Finally, solve the derived equation to express \( y \) in terms of \( x \).
### Interpolated Solution Box
Present the solution in a format similar to fill-in-the-blank, where the derived expression for \( y \) is typed using \( x \) as the variable.
For example:
\[ \text{The general solution is } y = \boxed{\frac{12}{x}} \]
(Note: The actual solution should be derived through proper step-by-step calculations specific to the given differential equation.)
### Graphs or Diagrams Explanation
There are no graphs or diagrams present in the provided text.
By following these steps, you can solve homogeneous differential equations systematically.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1887966a-d451-4216-a13a-dc9720678597%2F42f16ff2-9922-4ca0-877c-aad9e4cef962%2Fq3xf67d_processed.png&w=3840&q=75)
Transcribed Image Text:**Solving Homogeneous Differential Equations**
To solve the given differential equation using the method for homogeneous equations, follow these steps:
### Differential Equation
\[ (12y^2 - xy) \, dx + x^2 \, dy = 0 \]
### Task
Ignoring lost solutions, if any, the general solution is \( y = \) [___].
(Type an expression using \( x \) as the variable.)
### Explanation of Steps
1. **Rewrite the Differential Equation**: Express the differential equation in the standard form for homogeneous equations, if not already done.
2. **Substitution to Simplify**: Usually, substitute \( v = \frac{y}{x} \) and hence \( y = vx \). Consequently, \( dy = v \, dx + x \, dv \). Replace \( y \) and \( dy \) in the original equation.
3. **Separation of Variables**: After substituting, the equation should be separable. Separate the variables and integrate both sides to find the relationship between \( x \) and \( y \).
4. **Solve for \( y \)**: Finally, solve the derived equation to express \( y \) in terms of \( x \).
### Interpolated Solution Box
Present the solution in a format similar to fill-in-the-blank, where the derived expression for \( y \) is typed using \( x \) as the variable.
For example:
\[ \text{The general solution is } y = \boxed{\frac{12}{x}} \]
(Note: The actual solution should be derived through proper step-by-step calculations specific to the given differential equation.)
### Graphs or Diagrams Explanation
There are no graphs or diagrams present in the provided text.
By following these steps, you can solve homogeneous differential equations systematically.
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