y1,y2是一阶线性非齐次微分方程y'+p(x)y=Q(x)的两个特解,
所以,
y1'+p(x)y1=Q(x)
y2'+p(x)y2=Q(x)
λ,μ使λy1+μy2是该方程的解,
所以,
(λy1+μy2)'+p(x)(λy1+μy2)
=λ[y1'+p(x)y1]+μ[y1'+p(x)y1]
=λQ(x)+μQ(x)
=Q(x)
∴λ+μ=1
λy1-μy2是该方程对应的齐次方程的解,
所以,
(λy1-μy2)'+p(x)(λy1-μy2)
=λ[y1'+p(x)y1]-μ[y1'+p(x)y1]
=λQ(x)-μQ(x)
=0
∴λ-μ=0
∴λ=μ=1/2