Consider the differential equation
y′′+αy′+βy=t+e^(4t).
suppose the form of the particular solution to this differential equation as prescribed by the method of undetermined coefficients is
yp(t)=a1t^2+a0t+b0te^(4t).
determine the constants α and β.

Yp(t) = A1 t^2 + A0 t + B0 t e(4t)

=> y ' = 2A1t + A0 + B0 [e^(4t) +4 te^(4t) ]

     y ' = 2A1t + A0 + B0e^(4t) + 4B0 te^(4t)

=> y '' = 2A1 + 4B0e(4t) + 4B0 [ e^(4t) + 4te^(4t)

     y '' = 2A1 + 4B0e^(4t) + 4B0e^(4t) + 16B0te^(4t)

Now substitute the values of y ' and y '' in the differential equation:

 
y′′+αy′+βy=t+e^(4t)

 2A1 + 4B0e^(4t) + 4B0e^(4t) + 16B0te^(4t) + α{2A1t + A0 + B0e^(4t) + 4B0 te^(4t) } + β{A1 t^2 + A0 t + B0 t e(4t)} = t + e^(4t)

Next, we equate coefficients

1) Constant terms of the left side = constant terms of the right side:

2A1+ 2αA0 = 0 eq (1)

2) Coefficients of e^(4t) on both sides

8B0 + αB0 = 1 => B0 (8 + α) = 1 eq (2)

3) Coefficients on t

2αA1 + βA0 = 1 eq (3)

4) Coefficients on t^2

βA1 = 0 eq (4)

given that A1 ≠ 0 => β =0

5) terms on te^(4t)

16B0 + 4αB0 + βB0 = 0 => B0 (16 + 4α + β) = 0 ... eq (5)

Given that B0 ≠ 0 => 16 + 4α + β = 0

Use the value of β = 0 found previously

16 + 4α = 0 => α = - 16 / 4 = - 4.

α = - 4 and β = 0