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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 $f(W,t)$.

Itô's lemma expresses the differential of a function $f(W,t)$ as $df(W,t) = \left[f_t + \frac{1}{2} f_{WW}\right] dt + f_W dW$. This formula includes a second-order partial derivative term with respect to $W$ multiplied by $dt$, which arises from the non-zero quadratic variation of Brownian motion.

Conditions: $f$ is a sufficiently smooth function of $W$ and $t$.; $W$ is a standard Brownian motion.

Itô's lemma expresses the differential of a function $f(W,t)$ as $df(W,t) = \left[f_t + \frac{1}{2} f_{WW}\right] dt + f_W dW$. This formula includes a second-order partial derivative term with respect to $W$ multiplied by $dt$, which arises from the non-zero quadratic variation of Brownian motion.

Conditions: $f$ is a sufficiently smooth function of $W$ and $t$.; $W$ is a standard Brownian motion.