Final answer:
The differential equation in question can be rewritten in the standard differential form of M(x, y)dx + N(x, y)dy = 0, to facilitate finding a solution through common methods such as separation of variables or integrating factor.
Step-by-step explanation:
The differential equation y - 2y^4 = (y^7 + 3x)y' can be rewritten in differential form by bringing all terms involving y to one side and the terms involving x to the other, while separating the dx and dy parts.
This leads to an equation in the standard form of M(x, y)dx + N(x, y)dy = 0, where M and N are functions of x and y respectively. Upon doing this, we would obtain a differential equation that potentially could be solved using methods like separation of variables, integrating factor, or other methods depending on whether the equation is exact or can be made exact.
The given differential equation can be written in the form M(x,y)dx + N(x,y)dy = 0, where M(x,y) = y - 2y^4 and N(x,y) = (y^7 + 3x)y'. To transform the given differential equation into this form, you simply subtract 2y^4 from both sides of the equation. This allows us to identify the functions M(x,y) and N(x,y).