Second Order Equations-Reduction of Order: Problem 1

(1 point)

The differential equation

has as a solution.

Applying reduction order we set y = ux².

Then (using the prime notation for the derivatives)

y' =

=

So, plugging y into the left side of the differential equation, and reducing, we get

2².

d²y dy

dx²

-7x- + 16y=0

da

x²y" - 7xy + 16y=

The reduced form has a common factor of 5 which we can divide out of the equation so that we have cu" + u' = 0.

Since this equation does not have any u terms in it we can make the substitution wu' giving us the first order linear equation w' + w = 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 the general solution is y =