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It seems indeed that you hit a bug:

sage: var('x')
x
sage: function('y', x)
y(x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
TypeError:

When you write var('x'), two actions are done : it returns the symbolic variable (or "symbol") x and it gives to the Python variable (or 'name') x this value. This process of both returning a value and creating a Python variable with the same name is called variable injection, and it is a big source of confusion.

When you write var('x'), is is the same : it returns the symbolic function y(x) and it gives to the Python variable (or 'name') y this value.

The problem comes from the fact that function() does not inject its result into the Python variable y correctly:

sage: z = function('y', x)
sage: y
y
sage: z
y(x)
sage: type(y)
<class 'sage.symbolic.function_factory.NewSymbolicFunction'>
sage: type(z)
<type 'sage.symbolic.expression.Expression'>

As you can see, function('y', x) returned a correct symbolic expression y(x), while the Python variable y got some weird stuff.

So, the workaround is to overwrite the variable injection:

sage: x = var('x')
sage: y = function('y', x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
sage: de
x*e^y(x)*D[0](y)(x) == 2*x^3*e^(2*x) + 2*e^y(x)

Thanks for reporting, actually a similar bug was reported as trac ticket 15025.

It seems indeed that you hit a bug:

sage: var('x')
x
sage: function('y', x)
y(x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
TypeError:

When you write var('x'), two actions are done : it returns the symbolic variable (or "symbol") x and it gives to the Python variable (or 'name') x this value. This process of both returning a value and creating a Python variable with the same name is called variable injection, and it is a big source of confusion.

When you write var('x'), is is function('y', x), it is supposed to be the same : it returns should return the symbolic function y(x) and it gives give to the Python variable (or 'name') y this value.

The problem comes from the fact that function() does not inject its result into the Python variable y correctly:

sage: z = function('y', x)
sage: y
y
sage: z
y(x)
sage: type(y)
<class 'sage.symbolic.function_factory.NewSymbolicFunction'>
sage: type(z)
<type 'sage.symbolic.expression.Expression'>

As you can see, function('y', x) returned a correct symbolic expression y(x), while the Python variable y got some weird stuff.

So, the workaround is to overwrite the variable injection:

sage: x = var('x')
sage: y = function('y', x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
sage: de
x*e^y(x)*D[0](y)(x) == 2*x^3*e^(2*x) + 2*e^y(x)

Thanks for reporting, actually a similar bug was reported as trac ticket 15025.

It seems indeed that you hit a bug:

sage: var('x')
x
sage: function('y', x)
y(x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
TypeError:

When you write var('x'), two actions are done : it returns the symbolic variable (or "symbol") x and it gives to the Python variable (or 'name') x this value. This process of both returning a value and creating a Python variable with the same name is called variable injection, and it is a big source of confusion.

When you write function('y', x), it is supposed to be the same : it should return the symbolic function y(x) and it give to the Python variable (or 'name') y this value.

The problem comes from the fact that function() does not inject its result into the Python variable y correctly:

sage: z = function('y', x)
sage: y
y
sage: z
y(x)
sage: y == z
False
sage: type(y)
<class 'sage.symbolic.function_factory.NewSymbolicFunction'>
sage: type(z)
<type 'sage.symbolic.expression.Expression'>

As you can see, function('y', x) returned a correct symbolic expression y(x), while the Python variable y got some weird stuff.

So, the workaround is to overwrite the variable injection:

sage: x = var('x')
sage: y = function('y', x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
sage: de
x*e^y(x)*D[0](y)(x) == 2*x^3*e^(2*x) + 2*e^y(x)

Thanks for reporting, actually a similar bug was reported as trac ticket 15025.

It seems indeed that you hit a bug:

sage: var('x')
x
sage: function('y', x)
y(x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
TypeError:

When you write var('x'), two actions are done : it returns the symbolic variable (or "symbol") x and it gives to the Python variable (or 'name') x this value. This process of both returning a value and creating a Python variable with the same name is called variable injection, and it is a big source of confusion.

When you write function('y', x), it is supposed to be the same : it should return the symbolic function y(x) and it give to the Python variable (or 'name') y this value.

The problem comes from the fact that function() does not inject its result into the Python variable y correctly:

sage: z = function('y', x)
sage: y
y
sage: z
y(x)
sage: y == z
False
sage: type(y)
<class 'sage.symbolic.function_factory.NewSymbolicFunction'>
sage: type(z)
<type 'sage.symbolic.expression.Expression'>

As you can see, function('y', x) returned a correct symbolic expression y(x), while the Python variable y got some different stuff, more precisely a function (with no variable name specifies, somehow similar to cos or sin).

So, a first workaround is to overwrite the variable injection:

sage: x = var('x')
sage: y = function('y', x)
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
sage: de
x*e^y(x)*D[0](y)(x) == 2*x^3*e^(2*x) + 2*e^y(x)

Another workaround is to consider y as a function (whose variable name is not specified), and use the symbolic expression y(x), while the Python variable y got some weird stuff.

So, the workaround is to overwrite the variable injection: in your equation:

sage: x = var('x')
sage: y = function('y', x)
function('y')
y
sage: de = x*e^y*diff(y,x) == 2*(e^y+x^3*e^(2*x))
x*e^y(x)*diff(y(x),x) == 2*(e^y(x)+x^3*e^(2*x))
sage: de
x*e^y(x)*D[0](y)(x) == 2*x^3*e^(2*x) + 2*e^y(x)

Thanks for reporting, actually a similar bug was reported as trac ticket 15025.