First order non linear differential equation , non separable

Question

First time I run into an equation of this "form", for non linear equations I know separation of variables and substitution. It doesn't seem to be homogenous of degree $0$ so the substitution $v=\frac{x}{t}$ is out of the question? Applying $\ln$ to both sides didn't do much either

$$x'(t)=e^{t+x(t)}-1,\quad x(0)=1$$

Wolfram gives the solution $$x(t)=-\ln(c_1-t)-t$$

Answer 1

One can substitute $v(t)=t+x(t)$, which results in $v'(t)=1+x'(t)$ by differentiating both sides with respect to $t$. Hence, we have: $$x'(t)=e^{t+x(t)}-1\iff v'(t)-1=e^{v(t)}-1 \iff v'(t)=e^{v(t)}$$ This ODE is separable, and you know how to solve such.

Answer 2

Substituting $u(t)=t+x(t)$, we get $u'(t)-1=e^{u(t)}-1$ and $\frac{du(t)}{dt}=e^{u(t)}$. Now finish from here.

Answer 3

Read

$$x'(t)+1=(x(t)+t)'=e^{x(t)+t}$$

and immediately

$$e^{x(t)+t}=C-t.$$