Question #e964f

The problem is unsolvable as written. There is no exponent on the #3x# term.

The expression #(y+3x^)dy/dx=x # is not a proper expression for a differential equation. It is more likely that you meant to write either #(y+3x)^(dy/dx) = x#, or #(y+3x^n)dy/dx = x# (where #n# is whatever exponent you intended to include, e.g. 2, 3, etc), or possibly #(y+3x)^n dy/dx = x#

If you intended to write:

#(y+3x)^(dy/dx)=x#

Then there would indeed exist a solution, by taking #log_(y+3x)# of both sides:

#(y+3x)^(dy/dx) = x -> log_(y+3x) (y+3x)^(dy/dx)= log_(y+3x)x -> dy/dx = log_(y+3x)x#

If instead you intended to include an exponent #n#, such that the problem would be:

#(y+3x^n)dy/dx = x#

The solution would be found by dividing both sides by #(y+3x^n)#:

#(y+3x^n)dy/dx = x -> dy/dx = x/(y+3x^n)#

Where #n# is whatever exponent you intended to include.

Finally, if you intended to write:

#(y+3x)^n dy/dx = x#

You divide both sides by #(y+3x)^n# and obtain...

#(y+3x)^n dy/dx = x -> dy/dx = x/(y+3x)^n#

Where #n# is whatever exponent you intended to include.