Solve the differential equation dy/dx = (3x +2y)/ ( 3x + 2y + 2), using an appropriate substitution.

It's called a reduction to separation of variables.

dy/dx = (3x +2y)/ ( 3x + 2y + 2)

Let u = 3x + 2y

so

y = (1/2)*(u - 3x)

and

dy/dx = (1/2)*(du/dx - 3)

With this substitution, the equation becomes:

(1/2)*(du/dx - 3) = u/(u+2)

du/dx = 2u/(u+2) + 3 = (5u + 6)/(u+2)

This is a separable equation:

(u+2)/(5u+6) du = dx

Expand the left hand side in terms of partial fractions:

[1/5 - 4/(25u+30)] du = dx

Integrate:

[u/5 - (4/25)*ln(25u + 30)] = x - c

where c is the constant of integration.

Back substitute for u:

[(3x + 2y)/5 - (4/25)*ln(25(3x + 2y) + 30)] = x - c

This is an implicit solution for y. There is no explicit, closed-form solution that can be expressed in terms of elementary functions.