Question

(D³-5D²+8D-4)y=2e^x +e^2x​

Answer 1

Answer:

For the equation ( D^3 -5D^2 +8D -4)y = e^2x ,the characteristic equation is

m^3 -5m^2 +8m -4 = ( m-1)( m-2)^2 =0 , roots 1 , 2 ,2 .The complementary

function is yh = C1e^x + C2e^2x + C3xe^2x .

For the particular integral is appropriate the function yp = A(x^2)e^2x.

substituting in the equation gives A = 1/2 .Therefore the required solution is

y = C1e^x + C2e^2x + C3xe^2x +(1/2)(x^2)e^2x .

Answer 2

The required solution is y =  C₁(e^x) + C₂(e^2x) + C₃(xe^2x) +(1/2)(x^2)e^2x

For the equation ( D^3 -5D^2 +8D -4)y = e^2x ,

the characteristic equation is,

m^3 -5m^2 +8m -4 = ( m-1)( m-2)^2 =0

⇒ Roots are 1 , 2 ,2

The complementary function C.F. = C₁(e^x) + C₂(e^2x) + C₃(xe^2x)

For the particular integral  P.I. = A(x^2)e^2x.

substituting in the equation gives A = 1/2 .

Therefore the solution is

⇒ y = C₁(e^x) + C₂(e^2x) + C₃(xe^2x) +(1/2)(x^2)e^2x .

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