Consider $\displaystyle \dot x = x+1$. Find its solution x(t) satisfying the initial condition x(0)=0.
How should I attack this problem?
Remember that $\displaystyle \dot x =\frac{dx}{dt}$
This gives a first order seperable ODE.
$\displaystyle \frac{dx}{dt}=x+1 \iff \frac{dx}{x+1}=dt$
Integrating both sides we get
$\displaystyle \ln|x+1|=t+c$ solving for x gives
$\displaystyle x+1=Ae^{t} \iff x(t)=Ae^{t}-1$
From here just plug in your intial condition to find A.
Good luck.