= Solution
This is a <Bernoulli differential equation> with exponent four. On the nonzero branch selected by $y(0)=1$, put $v=y^{-3}$. Multiplying the original equation by $-3y^{-4}$ gives
$$
v'-v=-3e^x.
$$
Its <integrating factor> is $e^{-x}$, so $(e^{-x}v)'=-3$ and $e^{-x}v=C-3x$. The <initial condition> sets $C=1$. Thus
$$
\boxed{y(x)=\frac{e^{-x/3}}{(1-3x)^{1/3}},\qquad -\infty<x<\frac13}.
$$
This is the maximal real solution interval containing zero: the solution diverges as $x\to(1/3)^-$. The real negative branch of the same algebraic formula for $x>1/3$ is disconnected from this initial-value solution. The original equation also admits $y=0$, but it does not satisfy the prescribed <initial condition>.
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