If X=x+h, Y=y+k transforms d y d x = 2 x+3 y-7 3 x+2 y-8 to a homogeneous differential equation, then ( h , k )=

Step-by-step solution

d y d x = 2 x+3 y-7 3 x+2 y-8 ...(i) Let us use the transformation : aligned & X=x+h x=X-h & Y=y+k y=Y-k & And d X d Y = d x d y aligned Using above transformation in eq (i), we get aligned & d Y d X = 2(X-h)+3(Y-k)-7 3(X-h)+2(Y-k)-8 & d Y d X = 2 X+3 Y+(-2 h-3 k-7) 3 X+2 Y+(-3 h-2 k-8) aligned To make above differential equation homogeneous: -2 h-3 k-7=0 ...(ii) &-3 h-2 k-8=0 ...(iii) Solving equations (ii) & (iii), we gt aligned & h =2 & k =1 & ( ~h , k )=(2,1) aligned