Solve the first-order PDE using Lagrange’s method \[ p + q = x + y \] where \( p = \frac{\partial z}{\partial x} \) and \( q = \frac{\partial z}{\partial y} \).
Answer : #### **Solution:** 1. **Given PDE:** \[ p + q = x + y \] which can be rewritten as: \[ \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = x + y \] 2. **Using Lagrange's ... \( C_1 = x - y \), \[ z = y^2 + (x-y)y + C_2 \] which is the general solution....
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