1. Use Fourier series to solve the differential equation
$\frac{d^{2} y}{d t^{2}} + 25 y = f (t)$
where $f (t) = | t |, - π \leq t \leq π, f (t + 2 π) = f (t)$ .

Answer

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Answer

$y (t) = \sum_{n = 1}^{\infty} [\frac{- b_{n}}{25 - n^{2}} \cos (ω_{n} t) - \frac{a_{n}}{25 - n^{2}} \sin (ω_{n} t)]$

Steps

Step 1 ： $f (t) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} [a_{n} \cos (ω_{n} t) + b_{n} \sin (ω_{n} t)], ω_{n} = n$

Step 2 ： $a_{0} = \frac{2}{2 π} \int_{- π}^{π} | t | d t,$ $a_{n} = \frac{1}{π} \int_{- π}^{π} | t | \cos (n t) d t,$ $b_{n} = \frac{1}{π} \int_{- π}^{π} | t | \sin (n t) d t$

Step 3 ： $y (t) = \sum_{n = 1}^{\infty} [\frac{- b_{n}}{25 - n^{2}} \cos (ω_{n} t) - \frac{a_{n}}{25 - n^{2}} \sin (ω_{n} t)]$

1. Use Fourier series to solve the differential equationd2ydt2+25y=f(t)where f(t)=|t|,−π≤t≤π,f(t+2π)=f(t).

Answer

Popular Problems

1. Use Fourier series to solve the differential equation
$\frac{d^{2} y}{d t^{2}} + 25 y = f (t)$
where $f (t) = | t |, - π \leq t \leq π, f (t + 2 π) = f (t)$ .