please solve!

Question

x^2y^11-7x•y^1+16y=(lux+1)x^4

Answer 1

The given is transformed by x→e^t to a linear equation with constant coefficients: y”(t)-8y’(t)+16 y(t)=e^4t•(1+t)

Answer 2

Applying laplace transform y(t)→y(s) and solving for y(s) we obtain: y(s)= y(0)•s-8/(s-4)^2+y’(0)•1/(s-4)^2+1/(s-4)^3+1/(s-4)^4

Answer 3

For every term now we perform the inverse transform using the theorem of integral residues (cauchy)and after t→ lux,

Answer 4

L-1{y(0) s-8/(s-4)^2}= y(0).lim s→4 a^2/as[(s-8)e^st]= y(0)•e^4t•(1+4t)→ y(0)•x^4•(1+4 lux)

Answer 5

L-1{y'(0)•(s-4)^-2}=y'(0)•lim s→4 a^/as(e^st)=y'(0)•t•e^4t→ y'(0)•x^4•lux

Answer 6

L-1{(s-4)^-3}=1/2! lim s→4 a^2^/as^2(e^st)=t^2/2 e^4t→ x^4/2 lu^2x

Answer 7

L-1{(s-4)^-4}=1/3! lim s→4 a^3^/as^3(e^st)=t^3/6e^4t→x^4/6•lu^3x

Answer 8

Therefore the required solution is:

Answer 9

y(x)=x^4•[1/6 lu^3x+1/2lu^2x+(4y with index 0+ y' with inder 0)• lux+y with index 0)