The differential equation x22yx2−7xyx+16y=0
has x4 as a solution.
Applying reduction order we set y2=x4.
Then (using the prime notation for the derivatives)
y′2=
y′′2=
So, plugging y2 into the left side of the differential equation, and reducing, we get
x2y′′2−7xy′2+16y2=
The reduced form has a common factor of x5 which we can divide out of the equation so that we have x′′+′=0.
Since this equation does not have any u terms in it we can make the substitution =′ giving us the first order linear equation x′+=0.
This equation has integrating factor
for x > 0.
If we use a as the constant of integration, the solution to this equation is =
Integrating to get u, and using b as our second constant of integration we have =
Finally y2=
and the general solution is