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The equation $f_W = \sigma f$ is treated as an ordinary differential equation because it involves differentiation only with respect to $W$, and the variable $t$ does not appear in the derivative structure, acting merely as a parameter.

Conditions: $\sigma$ is a constant.; The equation is considered with respect to $W$ while $t$ is held fixed.

The equation $f_W = \sigma f$ is treated as an ordinary differential equation because it involves differentiation only with respect to $W$, and the variable $t$ does not appear in the derivative structure, acting merely as a parameter.

Conditions: $\sigma$ is a constant.; The equation is considered with respect to $W$ while $t$ is held fixed.

By substituting the ansatz $S=f(W,t)$ into the SDE and applying Itô's lemma, the coefficients of $dW$ and $dt$ are matched. This yields the system of PDEs: $f_W = \sigma f$ and $f_t + \frac{1}{2} f_{WW} = \mu f$.

Conditions: $S = f(W,t)$ is the assumed solution form.; $dS = \mu S\,dt + \sigma S\,dW$ is the defining SDE.; Itô's lemma is applied to $f(W,t)$.

By substituting the ansatz $S=f(W,t)$ into the SDE and applying Itô's lemma, the coefficients of $dW$ and $dt$ are matched. This yields the system of PDEs: $f_W = \sigma f$ and $f_t + \frac{1}{2} f_{WW} = \mu f$.

Conditions: $S = f(W,t)$ is the assumed solution form.; $dS = \mu S\,dt + \sigma S\,dW$ is the defining SDE.; Itô's lemma is applied to $f(W,t)$.