Question

solve the following ordinary differential equation for y(t) by using
the Laplace Transform technique.
y'' − 10y' + 9y = 5t (1)
with the initial conditions
y(0) = −1 and y'(0) = 2 (2)
(a) Apply the Laplace transform to (1) and solve for Y (s). solve the following ordinary differential equation for y(t) by using
the Laplace Transform technique.
y'' − 10y' + 9y = 5t (1)
with the initial conditions
y(0) = −1 and y'(0) = 2 (2)
(a) Apply the Laplace transform to (1) and solve for Y (s). @Mathematics

Answer 1

\[y''-10y'+9y=5t\quad;\quad y(0)=-1\quad,\quad y'(0)=2\]\[s^2Y-sy(0)-y'(0)-10(sY-y(0))+9Y=\frac{5}{s^2}\]\[s^2Y+s-2-10(sY+1)+9Y=\frac{5}{s^2}\]\[(s^2-10s+9)Y+s-12=\frac{5}{s^2}\]\[(s-1)(s-9)Y=\frac{5}{s^2}-s+12=\frac{-s^3+12s^2+5}{s^2}\]\[Y=\frac{-s^3+12s^2+5}{s^2(s-1)(s-9)}=\frac{50}{81s}+\frac{5}{9s^2}+\frac{-2}{s-1}+\frac{31}{81(s-9)}\]\[y=\frac{50}{81}+\frac{5}{9}t-2e^t+\frac{31}{81}e^{9t}\]