Problem 3 - Differential Equations and Linear Algebra

Question

Determine whether the given differential equation is exact. $$\left(y+3 x^{2}\right) d x+x d y=0$$

Step-by-Step Solution

Answer

The given differential equation can be expressed as \(M(x,y)dx + N(x,y)dy = 0\) with \(M(x, y) = y + 3x^2\) and \(N(x, y) = x\). After computing, we find that \(\frac{\partial M}{\partial y} = 1\) and \(\frac{\partial N}{\partial x} = 1\), which are equal. Therefore, the differential equation is exact.

Step 1: Identify M(x, y) and N(x, y)

From the given differential equation, we can identify the functions: \(M(x, y) = y + 3x^2\) and \(N(x, y) = x\). Step 2: Compute partial derivatives

Step 2: Compute partial derivatives

Now, we will compute the partial derivatives \(\frac{\partial M}{\partial y}\) and \(\frac{\partial N}{\partial x}\). \(\frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(y + 3x^2) = 1\) \(\frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(x) = 1\) Step 3: Compare partial derivatives

Step 3: Compare partial derivatives

Now, we compare the partial derivatives computed in Step 2. We find that: \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\) Since the partial derivatives are equal, the given differential equation is exact.

Key Concepts

Partial Derivatives

To understand exact differential equations, it's crucial to grasp the concept of partial derivatives. Partial derivatives help you see how a function of several variables changes with respect to one variable, while keeping the other variables constant.
For instance, if we have a function of two variables, say, \( f(x, y) = y + 3x^2 \), finding the partial derivative with respect to \( y \), noted as \( \frac{\partial f}{\partial y} \), means calculating how \( f(x, y) \) changes as \( y \) changes, assuming \( x \) remains fixed.

• This is helpful when you want to evaluate how the function behaves in multiple dimensions or spaces.
• It demonstrates how changing one dimension impacts the entire function.

Once you compute the partial derivatives for a given function, you can use them to solve differential equations, make predictions, and analyze complex phenomena.

M(x, y) and N(x, y)

Mapped within most differential equations are components, often marked as \( M(x, y) \) and \( N(x, y) \). These are notations for functions involved in an equation like \( M(x, y) dx + N(x, y) dy = 0 \). Here’s how to determine them:
In our example, \( M(x, y) = y + 3x^2 \) and \( N(x, y) = x \). This means:

• \( M(x, y) \) represents everything multiplied by \( dx \).
• \( N(x, y) \) represents everything multiplied by \( dy \).

By identifying these parts, you set the stage for checking a differential equation for exactness, which is essential for finding solutions.

Checking Exactness

The ultimate goal in dealing with differential equations like \( M(x, y) dx + N(x, y) dy = 0 \) is checking if they are exact. This process ensures that there's a potential function \( \Phi(x, y) \) that satisfies both functions within the equation. Here's how you do it:

• Compute the partial derivative \( \frac{\partial M}{\partial y} \).
• Compute the partial derivative \( \frac{\partial N}{\partial x} \).
• Compare both derivatives. If \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \), the equation is exact.

For our exercise, both partial derivatives equaled one: \( \frac{\partial M}{\partial y} = 1 \) and \( \frac{\partial N}{\partial x} = 1 \). This equality confirms the equation is exact, meaning we can solve it using methods suited for exact equations.