Question

partial differential equation by eliminating arbitrary functions z=f(x)+e^yg(x)​

Answer 1

Step-by-step explanation:

zx=f'(x)+e^yg(x)

zy=e^yg(x)

Answer 2

Explanation:

The method of eliminating arbitrary functions is used to convert a partial differential equation (PDE) with arbitrary functions into an equivalent PDE without arbitrary functions. The process involves substituting an expression for the arbitrary function into the PDE and then solving for the remaining unknown function.

In the example given, the PDE is given in the form z = f(x) + e^yg(x). To eliminate the arbitrary functions, we can substitute this expression into the PDE and solve for the remaining unknown function.

For example, if the PDE is given as:

$∂z/∂x + ∂z/∂y = 0$

We can substitute the given expression for z into the PDE:

$∂(f(x) + e^yg(x))/∂x + ∂(f(x) + e^yg(x))/∂y = 0$

Which simplifies to:

$f'(x) + g'(x)e^yg(x) + e^y(g'(x) + g(x)) = 0$

Now we can solve this equation for g(x) or f(x) by isolating them on one side of the equation and setting the other side to zero.

It should be noted that this method can be applied to any PDE with arbitrary functions, but the complexity of the solution will vary depending on the form of the PDE and the expressions for the arbitrary functions.

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