What general form does the first PDE force for $f(W,t)$?

Conditions

• $f_W = \sigma f$ holds.
• $\sigma$ is constant.

Reasoning, step by step

1. Start with the ODE $f_W = \sigma f$ treating $t$ as a parameter.
2. Integrate with respect to $W$ to get $f(W,t) = C e^{\sigma W}$.
3. Allow the integration constant $C$ to depend on $t$, denoting it as $\varphi(t)$.
4. Conclude the separated form $f(W,t) = \varphi(t)e^{\sigma W}$.

Example

The board writes '$f(W,t) = \varphi(t) e^{\sigma W}$' after solving the first PDE.

Common misconceptions

• Assuming the integration constant is a true constant independent of $t$.
• Forgetting the exponential dependence on $W$.

Watch the explanation

Explore next

• What derivatives are computed next in order to use the second PDE?
• Why does solving $f_W=\sigma f$ introduce a function $\phi(t)$ rather than a constant?