2.1.2.5 ✓ Mathematica. Time used: 0.005 (sec). Leaf size: 14

ode=D[x[t],t]==b*Exp[t]; 
ic={x[1]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to b \left (e^t-e\right ) \end{align*}