Verifying a Solution for a First-Order Partial Differential Equation

Posted by on Oct 3, 2025 in Mathematical Engineering | 0 comments

Problem Statement and Given Solution

We are tasked with showing that the given function $u$ satisfies the following partial differential equation (PDE):

𝑥𝜕𝑢𝜕𝑥−𝑦𝜕𝑢𝜕𝑦=𝑦2𝑢3

The proposed solution is:

𝑢=(1+2𝑥𝑦+𝑦2)−12

1. Calculate the Partial Derivative 𝜕𝑢/𝜕𝑥

Using the chain rule, we differentiate $u$ with respect to $x$:

𝜕𝑢𝜕𝑥=−12(1+2𝑥𝑦+𝑦2)−32⋅(2𝑦)=−𝑦(1+2𝑥𝑦+𝑦2)−32

2. Calculate the Partial Derivative 𝜕𝑢/

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