Let g(x)=py 1 +2qy2, and bring g(x) into y'+Py =0.

py 1 '+2q y2 '+PPy 1+2q py 2 = 0-( 1)

Because y 1'+py 1 = q, y2'+py2 = q.

So py 1'+pPy 1 = pQ.

2qy2'+2qPy2 = 2qQ

Their sum is pQ(x)+2qQ(x) =0, so p+2q=0.

Similarly, we can get p-q = 1.