Solving the Partial Differential Equation ∂z/∂t = z - 1

Solving the Partial Differential Equation ∂z/∂t = z - 1

This article explains how to solve the partial differential equation ∂z/∂t = z - 1 for z as a function of t.

1. Separation of Variables

Begin by rearranging the equation to separate the variables z and t:

∂z/(z - 1) = dt

2. Integration

Integrate both sides of the equation. The left side is integrated with respect to z, and the right side with respect to t:

∫∂z/(z - 1) = ∫dt

This gives us:

ln|z - 1| = t + C

where C is the constant of integration.

3. Solving for z

Solve the equation for z:

|z - 1| = e^(t + C)

Since the absolute value of an expression is always non-negative, we can remove the absolute value and introduce a new constant, k, which can be positive or negative:

z - 1 = k * e^(t)

Finally, we get the general solution for z:

z = 1 + k * e^(t)

Solving for z in terms of t and s

The provided equation only relates z and t. To find a specific solution for z in terms of both t and s, we would need additional information, such as another equation or boundary conditions, that establish a relationship between t and s.