Question

Solve the differential equation with details explaination :

x²y" + 6xy' - 24y=x^9

Answer 1

let, y= x^r

or, y' = ry^r-1

or, y"= r(r-1) x^r-2

Now from the given differential equation in homogenous form we have

r(r-1)x^r+6rx^r-24x^r= 0

or, r²-r+6r-24=0

or, r²+5r-24=0

or, (r-3)(r+8)=0

or, r=3,-8

The complementary function is

C.F= C1x^3+C2x^-8

Now for the particular solution we have

P.I = x^9/(9²+5×9-24)= x^9/102

Hence y = C1x^3+C2^x-8+x^9/102

 The complementary function is

C.F= C1x3+C2x-8

Now for the particular solution we have

P.I = x^9/(9²+5×9-24)= x^9/102

Hence y = C1x^3+C2^x-8+x^9/102