I was recently given a question to solve by my professor that required me to find the Inverse Laplace. I'm sure my solution was correct but I'm wondering if my professors solution is also correct.
My professor and I both manage to start off the problem the same way but as soon as we get to partial fractions our work begins to differ. I want to include that I do understand what my professor did to get the answer he did. He used the formula for partial fraction expansion and inverse shown below. I just want to know if both answers are somehow equivalent to one another because I'm having trouble seeing whether or not they are.
The question: Solve the differential equation:
d^2(y(t))/d^2(t) + 8d(y(t))/d(t) + 20y(t) = 10e^-2t u(t). Assume y(0)=y'(0)=0
My Solution:
S^2Y(s)+8SY(s)+20Y(s) = 10/s+2
Y(s)=10/(s+2)(S^2+8s+20)
Taking the partial fraction(This is where my professor and I begin to differ):
A/s+2 + Bs+c/s^2+8s+20
10 = A(s^2+8s+20) + Bs+C(s+2)
s^2(A+B) + s(8A+2B+C) + 20A+2C
A=5/4, B=-5/4, C=-15/2
Solution:
y(s) = 5/4 × (1/s+2) - (5/4)(s+4) × 1/((s+4)^2+2^2) - 5/4 × 2/((s+4)^2+2^2)
Inverse Laplace:
y(t) = (5/4 × e^(-2t) - 5/4 × e^(-4t)cos(2t) - 5/4 × e^(-4t)sin(2t)) u(t)
Professors Solution:
partial fractions:
10/(s+2)((s+4)^2 + 2^2) = A/(s+2) + B/(s+4+2j) + B(conjugate)/(s+4-2j)
A = 1/4, B = (10/8√2) × (e^(jπ/4)), B(conjugate) = (10/8√2) × (e^(-jπ/4))
Solution:
Y(s) = (1/4)/(s+2) + (10/8√2)(e^(jπ/4))/(s+4+2j) + (10/8√2)(e^(-jπ/4))/(s+4-2j)
Inverse Laplace:
y(t) = [1/4e^(-2t) + (10/8√2)(e^(-4t))cos(2t+π/4)] u(t)
I Understand that my professor used the formula for partial fraction expansion and inverse when solving it his way which in this case states when Y(s) = B/(s+4+2j) + B(conjugate)/(s+4-2j) then, y(t) = 2|B|e^(-4t)cos(2t+π/4)u(t)
I understand how to solve the problem both ways but I don't know if both answers are equivalent and if so, how? Which method is preferred or should you just go with whichever method is preferred? Thanks.
Edit: I thought I'd include pictures of my work and the professors work if it made it easier.
Partial Fraction Expansion and Inverse
2nd Edit: I think my professor made some mistakes in his solution so now I'm even more confused.