Final answer:
The solution to the given first-order linear PDE is z = xp + yq, making option (b) z = xp + yq the correct answer. The PDE is solved by identifying z as a linear combination of x and y.
Step-by-step explanation:
The given partial differential equation (PDE) is: xp + yq = z. Here, 'p' and 'q' are partial derivatives of z with respect to x and y, respectively. To solve this PDE, we recognize it as a first-order linear PDE.
We are searching for a function z(x,y) such that the partial derivatives of z with respect to x and y yield an expression that fits the given PDE. Based on the form of the equation, we can see that z is a linear combination of x and y, motivated by the terms xp and yq.
By integrating 'p' with respect to 'x', we get 'zx = x + f(y)', where f(y) is an arbitrary function of 'y'. Similarly, integrating 'q' with respect to 'y' gives us 'zy = y + g(x)', where g(x) is an arbitrary function of 'x'. Comparing these to the original equation, we see that z can be rewritten as z = xp + yq. Therefore, the correct answer aligns perfectly with option (b).