Partial Differential Equations Waves in a String

SOLUTION We convert the equation to the pfaffian form, y dx — x dy = 0. Test for exactness  

The equation is not exact. We show that 1 jx is an integrating factor. Multiplication by this factor 

This is a general solution, sinee the original equation was first order and the solution contains one 

If an inexact differential has one integrating factor, it has an infinite number of integrating factors. Therefore, there can be other integrating factors for a differential such as the one in the preceding example. Unfortunately, there is no general procedure for finding an integrating factor except by trial and error. 

Show that 1/y and / x + y ) are integrating factors for the equation in the previous example and show that they lead to the same solution. Q 

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Section 8.6 Partial Differential Equations Waves in a String... 

The differential equation of the vibrating string is the one-dimensional wave equation, which is a partial differential equation, second order in each of t and x ... 

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