Urgent help for math differential equations

Question

Consider the equation y′′′ + y′′ = 3e^x+ 4x^2.

(a) Is yc = c1 + c2x + c3e^x or yc = c1 + c2x + c3e^−x complimentary solutions to the homogeneous equation associated with the given non-homogeneous equation ?

(b) Is yp = 3/2e^x + 4x^2 − 4/3x^3 + 1/3x^4 a particular solution to the given non-homogeneous equation ?

Answer 1

y''' + y'' = 3e^x + 4x²;

Characteristics equation: r³ + r² = 0; r²(r + 1) = 0 r = 0 multiplicity 2 or r = -1

(a)Then general solution of homogeneous equation y_c = C₁ + C₂x + C₃e^-x corresponding to this ODE

(b) y_p is particular solution of this of this ODE you can check by substitution