Let `y_(1)` and `y_(2)` be two different solutions of the equation
`(dy)/(dx)+P(x).y=Q(x)`. Then `alphay_(1)+betay_(2)` will be solution of the given equation if `alpha + beta=……………….`

Answer

1

Explanation

Given that `y_(1)` and `y_(2)` are two different solutions of the differential equation.
`(dy)/(dx) + P(x).y=Q(x)`
`rArr (dy_(1))/(dx)+ P(x).y_(1)=Q(x)`
and `(dy_(2))/(dx) + P(x).y_(2)=Q(x)`
Now, `ay_(1)+betay_(2)` will be a solution
If `d/(dx)(ay_(1)+betay_(2))+P(x).(ay_(1)+betay_(2))=Q(x)`
`rArr a{(dy_(1))/(dx) +P(x) .y_(1)}+beta{(dy_(2))/(dx)+P(x).y_(2)}=Q(x)`
`rArr aQ(x) + betaQ(x)=Q(x)`
`rArr alpha+beta=1`