maths -
math notes for differential: solve with undermined coefficients.
question 9, 4.5.21
solve y''(θ)+12y'(θ)+37y(θ)=Power[e,-6θ]cosθ for θ
question 10, 4.5.25
{z''(x) + z(x) = 9 e^(-6 x), z(0) = 0, z'(0) = 0}
question 15, 4.5.43
Given the general form for the differenrial equation for the spring-mass system: $$ my''+by'+ky = g(t) $$ and $$ mass: m = 1, spring-constant: k = 6,damping-coefficient: b=5, start: y(0) = \frac{1}{2}, rest: y'(0) = 0. $$
Then solve above for
solve y''(t)+5y'(t)+6y(t)=6*sin(2t) for t
The above gives:
y(t) = c_1 e^(-3 t) + c_2 e^(-2 t) + 3/26 sin(2 t) - 15/26 cos(2 t)
e.i.
$$
y(t) = c_1 e^{-3t} + c_2 e^{-2 t} + \frac{3}{26}sin(2t) - \frac{15}{26} cos(2t)
$$
Evaluate
$$
\implies y(0) = c_1 + c_2 - \frac{15}{26}
$$
and
$$
y(0)=\frac{1}{2}
$$
$$
\implies
c_1 + c_2 - \frac{15}{26} = 0.
$$
Differiate
derevative y(t) = c_1 Power[e,\(40)-3 t\(41)] + c_2 Power[e,\(40)-2 t\(41)] + Divide[3,26] sin(2 t) - Divide[15,26] cos(2 t) where t=0
$$
y'(0)= -3 c_1 - 2 c_2 + \frac{3}{13}
$$
and
$$
y'(0) = 0
$$
$$
\implies
-3 c_1 - 2 c_2 + \frac{3}{13} = 0.
$$
Solve system of equations to find