if the equation (2ax^2+2 e^y)dx+(2xe^y+3y^2)dy is exact. Find the value of a and b.

To determine if the differential equation 

\[
(2ax^2 + 2e^y) \, dx + (2xe^y + 3y^2) \, dy = 0
\]

is exact, we need to check if the equation satisfies the condition for exactness. For a differential equation of the form \( M(x, y) \, dx + N(x, y) \, dy = 0 \), it is exact if:

\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]

Here, \( M(x, y) = 2ax^2 + 2e^y \) and \( N(x, y) = 2xe^y + 3y^2 \).

Let's compute the partial derivatives:

1. **Partial derivative of \( M(x, y) \) with respect to \( y \)**:
   \[
   \frac{\partial M}{\partial y} = \frac{\partial}{\partial y} (2ax^2 + 2e^y) = 0 + 2e^y = 2e^y
   \]

2. **Partial derivative of \( N(x, y) \) with respect to \( x \)**:
   \[
   \frac{\partial N}{\partial x} = \frac{\partial}{\partial x} (2xe^y + 3y^2) = 2e^y
   \]

For the equation to be exact, we need \(\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\). Since both partial derivatives are equal:

\[
2e^y = 2e^y
\]

The equation is indeed exact, and no additional conditions on \( a \) and \( b \) are required for this exactness, other than what is given.

Therefore, the value of \( a \) can be any real number. There is no \( b \) specified in the problem, so no value for \( b \) needs to be determined.

Would you like a further explanation or have any questions?

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Here are 5 related questions:

1. How can you solve an exact differential equation?
2. What does it mean for a differential equation to be "exact"?
3. What are the steps to find a potential function for an exact differential equation?
4. How do you apply the method of integrating factors to make a non-exact equation exact?
5. Can exactness be generalized to higher-order differential equations?

**Tip:** Exact differential equations can often be solved by finding a potential function \( \psi(x, y) \) such that \( \frac{\partial \psi}{\partial x} = M(x, y) \) and \( \frac{\partial \psi}{\partial y} = N(x, y) \).

Explore the exactness of the differential equation (2ax^2 + 2 e^y)dx + (2xe^y + 3y^2)dy and find the values of a and b. Learn about the condition for exactness in differential equations with step-by-step solutions. Suitable for advanced undergraduate students and anyone interested in exact differential equations.

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