Question: The differential equation dy dy 7x + 16y = 0 da? has a* as a solution. Applying reduction order we set y2 = ur

The differential equation dy dy 7x + 16y = 0 da? has

The differential equation dy dy 7x + 16y = 0 da? has a* as a solution. Applying reduction order we set y2 = ur Then (using the prime notation for the derivatives) = So, plugging y2 into the left side of the differential equation, and reducing, we get r'y 7ry + 16y = The reduced form has a common factor of 25 which we can divide out of the equation so that we have ru" + u' = 0. Since this equation does not have any u terms in it we can make the substitution w = u' giving us the first order linear equation rw' + w = 0. This equation has integrating factor for x > 0. If we use a as the constant of integration, the solution to this equation is w = Integrating to get u, and using b as our second constant of integration we have u = Finally y2 = and the general solution is

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