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When solving $f_W = \sigma f$ with respect to $W$, the variable $t$ is treated as a parameter. Therefore, the 'constant' of integration can be any function of $t$, denoted as $\varphi(t)$.

Conditions: The equation is solved with respect to $W$.; $t$ is held fixed during the integration.

When solving $f_W = \sigma f$ with respect to $W$, the variable $t$ is treated as a parameter. Therefore, the 'constant' of integration can be any function of $t$, denoted as $\varphi(t)$.

Conditions: The equation is solved with respect to $W$.; $t$ is held fixed during the integration.

Solving the first PDE $f_W = \sigma f$ forces the general form $f(W,t) = \varphi(t)e^{\sigma W}$, where $\varphi(t)$ is an arbitrary function of time.

Conditions: $f_W = \sigma f$ holds.; $\sigma$ is constant.

Solving the first PDE $f_W = \sigma f$ forces the general form $f(W,t) = \varphi(t)e^{\sigma W}$, where $\varphi(t)$ is an arbitrary function of time.

Conditions: $f_W = \sigma f$ holds.; $\sigma$ is constant.