find the plane equation of the ODE system x’ = -2x+3y and y’=4x-6y

The given system of ordinary differential equations is:

\[
x' = -2x + 3y
\]
\[
y' = 4x - 6y
\]

We aim to find the plane equation that represents this system. To do this, we treat it as a linear system of differential equations, and we seek an implicit relationship between \(x\) and \(y\) by eliminating the time derivatives.

### Step 1: Find the derivatives
Let’s differentiate \(x' = -2x + 3y\) with respect to time \(t\):

\[
x'' = -2x' + 3y'
\]

Substitute \(x'\) and \(y'\) from the original system:
\[
x'' = -2(-2x + 3y) + 3(4x - 6y)
\]

### Step 2: Simplify the equation
Now, simplify the terms:

\[
x'' = 4x - 6y + 12x - 18y
\]
\[
x'' = 16x - 24y
\]

### Step 3: General form of the solution
This second-order differential equation represents the system in terms of \(x\) and \(y\). The equation \(16x - 24y = 0\) can be simplified to:

\[
\frac{x}{3} = \frac{y}{2}
\]

Thus, the equation of the plane for this system of differential equations is:

\[
2x = 3y
\]

This is the linear relationship between \(x\) and \(y\), which can be interpreted as the equation of the plane.

---

Would you like more details or have any questions?

Here are five related questions to deepen understanding:
1. How would the solution look for a nonlinear system of ODEs?
2. What role does the Jacobian matrix play in analyzing systems of differential equations?
3. How do you find the general solution to this system of differential equations?
4. Can this system of ODEs be expressed in matrix form? If so, what would it look like?
5. How would eigenvalues and eigenvectors help in solving a linear ODE system?

**Tip:** When solving a system of linear ODEs, always check for opportunities to use matrix methods, such as diagonalization, to simplify the process.

Learn how to find the plane equation from the ODE system x' = -2x + 3y and y' = 4x - 6y. This guide covers the step-by-step process, including calculating second-order derivatives and simplifying linear relationships. Suitable for advanced high school and undergraduate 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