I need help with this please I would appreciate your help Fi

I need help with this please. I would appreciate your help!

Find the form of a particular solution to y\" + 3\' - 3y\' = x^4e^x. Enter your solution as y_p(x) =.... In your answer, use - a_0 - a_4 denote arbitrary constants and x the independent variable. Enter as a_0 and a-0 as a_1, as a_1, etc.. Answer: y_p(x) =

Solution

Given that

y\'\' + 3y\' - 7y = x⁴e^x

D-operator form is ,

   ( D₂ + 3D -7 )y =x⁴e^x

Auxialary euation is,

m² + 3m - 7 = 0

This equation is in the form of ax² + bx + c = 0

a = 1, b = 3 , c = -7

m = [ -b ± (b² - 4ac) ] / 2a

= [ -3 ± (3² - 4.1.(-7)) ] / 2.1 ]

= [ -3 ± (37) ] / 2 ]

Hence ,

m₁ =  [ -3 + (37) ] / 2 , m₂ =  [ -3 - (37) ] / 2

Complementary function y_c = c₁e ^{[( -3 + (37) ) / 2]x} + c₂^{[( -3 - (37) ) / 2]x}

For a non homogeneous part x⁴e^x we assume the perticular solution of the form is ,

y_p(x) = ( a₀x⁴ + a₁x³ + a₂x² + a₃x + a₄ )e^x