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  

https://www.wolframalpha.com/input/?i2d=true&i=solve+y%27%27%5C%2840%29%CE%B8%5C%2841%29%2B12y%27%5C%2840%29%CE%B8%5C%2841%29%2B37y%5C%2840%29%CE%B8%5C%2841%29%3DPower%5Be%2C-6%CE%B8%5Dcos%CE%B8+for+%CE%B8++

question 10, 4.5.25

{z''(x) + z(x) = 9 e^(-6 x), z(0) = 0, z'(0) = 0}

https://www.wolframalpha.com/input/?i2d=true&i=z%27%27%5C%2840%29x%5C%2841%29+%2B+z%5C%2840%29x%5C%2841%29+%3D+9Power%5Be%2C%5C%2840%29-6x%5C%2841%29%5D%5C%2844%29+z%5C%2840%290%5C%2841%29%3D0%5C%2844%29+z%27%5C%2840%290%5C%2841%29%3D0

question 15, 4.5.43

Given the general form for the differenrial equation for the spring-mass system: $$ my''+by'+ky = g(t) $$ and $$ mass: m = 1, spring-constant: k = 6,damping-coefficient: b=5, start: y(0) = \frac{1}{2}, rest: y'(0) = 0. $$

$$ \implies y''+5y'+6y=6sin(2t) $$

Then solve above for $t$:

solve y''(t)+5y'(t)+6y(t)=6*sin(2t) for t

https://www.wolframalpha.com/input/?i2d=true&i=solve+y%27%27%5C%2840%29t%5C%2841%29%2B5y%27%5C%2840%29t%5C%2841%29%2B6y%5C%2840%29t%5C%2841%29%3D6*sin%5C%2840%292t%5C%2841%29+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 $y(t)$ for $t=0$

$$ \implies y(0) = c_1 + c_2 - \frac{15}{26} $$ and $$ y(0)=\frac{1}{2} $$ $$ \implies c_1 + c_2 - \frac{15}{26} = 0. $$ Differiate $y(t)$ and evaluate $y'(t)$ for $t=0$

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

https://www.wolframalpha.com/input/?i2d=true&i=derevative+y%5C%2840%29t%5C%2841%29+%3D+c_1+Power%5Be%2C%5C%2840%29-3+t%5C%2841%29%5D+%2B+c_2+Power%5Be%2C%5C%2840%29-2+t%5C%2841%29%5D+%2B+Divide%5B3%2C26%5D+sin%5C%2840%292+t%5C%2841%29+-+Divide%5B15%2C26%5D+cos%5C%2840%292+t%5C%2841%29+where+t%3D0

$$ y'(0)= -3 c_1 - 2 c_2 + \frac{3}{13} $$ and $$ y'(0) = 0 $$ $$ \implies -3 c_1 - 2 c_2 + \frac{3}{13} = 0. $$ Solve system of equations to find $c_1$ and $c_2$: