Question about Differential Equations, why is there no constant for $x$?

Question

The following equation is in the form$$\frac{dy}{dx}=f(y)$$ Solve the diffrential equation$$\frac{dy}{dx}=2+3y$$

$$\frac{dy}{dx}=2+3y$$ $$dy=(2+3y)dx$$ $$\frac{dy}{2+3y}=dx$$

then I integrated bothsides.. $$\int\frac{dy}{2+3y}=\int{dx}$$

and this is what I think you should get but the book says otherwise... $$\frac{1}{3}ln(2+3y)+c = x+c$$

I know I'm wrong about this but why is there no constant when we integrate the RHS? I'm assuming there's a 1 in front of the $dx$ and when you integrate you get $x+c$

If someone could explain why I'd be grateful.

Answer 1

The two sides can take different constants, say $c_1$ and $c_2$. But then you can move both constants to one side and call $c = c_1 - c_2$, which also can take any value.

The lesson here is that $c_1$ and $c_2$ can be same or different, in contrary to what your form says.

Answer 2

Formally speaking, this solution should be $$\frac{1}{3}\ln(2+3y)+c_1=x+c_2.$$ But we can also write this $$\frac{1}{3}\ln(2+3y)=x+c_2-c_1.$$ But we can define a new constant $c=c_2-c_1$, and write this $$\frac{1}{3}\ln(2+3y)=x+c.$$ Usually, we skip all the middle steps and jump straight to the end.