The given initial value problem is:

$y^3 - 5x^4 - 3xy^2 + e^x = \left( 3x^2 y - 3xy^2 + y^2 + \cos y \right) y'$

with the initial condition $y(0) = \frac{\pi}{2}$.

Step 1: Rearrange the equation

We start by rearranging the equation to separate the terms involving $y'$:

$y^3 - 5x^4 - 3xy^2 + e^x = \left( 3x^2 y - 3xy^2 + y^2 + \cos y \right) y'$

Now, divide both sides by $3x^2 y - 3xy^2 + y^2 + \cos y$:

$y' = \frac{y^3 - 5x^4 - 3xy^2 + e^x}{3x^2 y - 3xy^2 + y^2 + \cos y}$

Step 2: Integrate both sides

To solve for $y(x)$ implicitly, we need to integrate both sides with respect to $x$:

$\int \frac{dy}{y^3 - 5x^4 - 3xy^2 + e^x} = \int \frac{dx}{3x^2 y - 3xy^2 + y^2 + \cos y}$

However, this integral is highly non-trivial and not easily solvable by elementary functions. Instead, we use the given initial condition to write the implicit solution.

Step 3: Use the initial condition

Given $y(0) = \frac{\pi}{2}$, the solution is implicitly defined by a function $F(x, y) = C$ where $C$ is a constant determined by the initial condition.

The relation defining $y$ implicitly as a function of $x$ is given by:

$F(x, y) = \int \frac{dy}{y^3 - 5x^4 - 3xy^2 + e^x} = C$

Substituting $x = 0$ and $y = \frac{\pi}{2}$:

$F(0, \frac{\pi}{2}) = C$

Thus, the solution is given implicitly by the relation:

$\int \frac{dy}{y^3 - 5x^4 - 3xy^2 + e^x} = \int \frac{dx}{3x^2 y - 3xy^2 + y^2 + \cos y} + C$

Where $C$ is determined by the initial condition $F(0, \frac{\pi}{2})$.

This is the implicit form of the solution for the initial value problem.

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Would you like further details or have any questions?

Here are some related questions:

1. How do you solve implicit differential equations in general?
2. What are the common methods for solving first-order differential equations?
3. How do initial conditions affect the solution of differential equations?
4. What are some numerical methods for solving differential equations when analytical solutions are complex?
5. How do you interpret the results of an implicit solution?
6. What is the significance of the constant of integration in differential equations?
7. What techniques can simplify complex integrals in differential equations?
8. Can differential equations always be solved explicitly?

Tip: For complex differential equations that are difficult to solve analytically, consider numerical methods like Euler's or Runge-Kutta.