The general solution of the differential equation d y d x = x+7 y+3 3 x+5 y+9 is

Step-by-step solution

We have, d y d x = x+7 y+3 3 x+5 y+9 Put, x=x+h and y=y+k d y d x = (x+7 y)+(h+7 k+3) (3 x+5 y)+(3 h+5 k+9) Put h+7 k+3=0 and 3 h+5 k+9=0 h=-3, k=0 d y d x = x+7 y 3 x+5 y Now, put y=v x d y d x =v+x d v d x x d v d x = 1+7 v-3 v-5 v^2 3+5 v aligned & x d v d x =- [ 5 v^2-4 v-1 5 v+3 ] & 5 v+3 5 v^2-4 v-1 d v= - 1 x d x & 1 2 10 v+6 5 v^2-4 v-1 d v=- 1 x d x & 1 2 10 v-4+10 5 v^2-4 v-1 d v=- 1 x d x & 1 2 10 v-4 5 v^2-4 v-1 d v+5 1 5 v^2-4 v-1 d v & =- x+ k & 1 2 [5 v^2-4 v-1 ]+ 5 5 1 v^2- 4 5 v- 1 5 d v= ( k x ) &