Solue theDE by Excat method:- (x^3 +y^3)dx +3xy^2 +dy=0Solue theDE by Excat method:- (x^3 +y^3)dx +3xy^2 +dy=0

To solve the differential equation by the exact method, we need to verify that the given differential equation is exact and then find the solution.

### Given Differential Equation:

\[
(x^3 + y^3) \, dx + (3xy^2 + 1) \, dy = 0
\]

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

The equation can be written in the standard form:

\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]

where:

\[
M(x, y) = x^3 + y^3
\]

\[
N(x, y) = 3xy^2 + 1
\]

### Step 2: Check for Exactness

To check if the equation is exact, compute the partial derivatives:

\[
\frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (x^3 + y^3) = 3y^2
\]

\[
\frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (3xy^2 + 1) = 3y^2
\]

Since \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\), the differential equation is exact.

### Step 3: Find the Potential Function \(\psi(x, y)\)

Since the equation is exact, there exists a potential function \(\psi(x, y)\) such that:

\[
\frac{\partial \psi}{\partial x} = M(x, y)
\]

\[
\frac{\partial \psi}{\partial y} = N(x, y)
\]

First, integrate \(M(x, y)\) with respect to \(x\):

\[
\psi(x, y) = \int (x^3 + y^3) \, dx
\]

\[
\psi(x, y) = \frac{x^4}{4} + y^3x + h(y)
\]

where \(h(y)\) is a function of \(y\) alone.

Next, differentiate \(\psi(x, y)\) with respect to \(y\) and equate it to \(N(x, y)\):

\[
\frac{\partial \psi}{\partial y} = 3y^2x + h'(y)
\]

Equating this to \(N(x, y)\):

\[
3y^2x + h'(y) = 3xy^2 + 1
\]

Thus,

\[
h'(y) = 1
\]

Integrating with respect to \(y\):

\[
h(y) = y + C
\]

### Step 4: Write the General Solution

Substituting \(h(y)\) back into \(\psi(x, y)\), the potential function becomes:

\[
\psi(x, y) = \frac{x^4}{4} + y^3x + y + C
\]

The general solution is given by:

\[
\psi(x, y) = C
\]

So, the solution to the differential equation is:

\[
\frac{x^4}{4} + y^3x + y = C
\]

This is the required general solution.

Would you like more details or have any questions?

Here are five related questions:

1. How do you determine if a differential equation is exact?
2. What are the next steps if the equation is not exact?
3. Can you derive the potential function \(\psi(x, y)\) in a non-exact differential equation?
4. What is the physical interpretation of an exact differential equation?
5. How does the solution \(\psi(x, y) = C\) relate to level curves in a geometric sense?

**Tip:** When integrating to find the potential function, ensure you include the constant of integration properly to capture all possible solutions.

Explore the solution to the exact differential equation (x^3 + y^3)dx + 3xy^2 + dy = 0 using the exact method. Understand the integration techniques involved and learn how to verify exactness. Suitable for advanced undergraduate students and those interested in differential equations.

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