ODE: $y'=3e^{2(x+y)}-1$, $y(0)=7$

Question

How can I solve the following differential equation? $$y'=3e^{2(x+y)}-1,\quad y(0)=7$$

I am failing to separate the variables, and I am not yet introduced to other solving-methods.

Thanks in advance.

Answer 1

Call $u(x) = y(x) + x$ and note that

$$ \frac{{\rm d}u}{{\rm d}x} = \frac{{\rm d}y}{{\rm d}x} + 1 $$

So that your equation becomes

$$ \frac{{\rm d}y}{{\rm d}x} + 1 = 3e^{2(x+y)} ~~~\Rightarrow \frac{{\rm d}u}{{\rm d}x} = 3e^{2u} $$

Can you take it from here?