Ecuación diferencial lineal

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Ecuación diferencial lineal

  1. 1. Ecuación diferencial lineal.<br />Factor integrante.<br />
  2. 2. Separamos / dividimos:<br />𝑑𝑦𝑑𝑥+𝑐𝑜𝑠3𝑥𝑐𝑜𝑠2𝑥 𝑠𝑒𝑛𝑥𝑦=1𝑐𝑜𝑠2𝑥 𝑠𝑒𝑛𝑥<br />y’+𝑐𝑜𝑠𝑥𝑠𝑒𝑛𝑥𝑦=1𝑐𝑜𝑠2𝑥 𝑠𝑒𝑛 𝑥𝑝𝑥=cos𝑥𝑠𝑒𝑛 𝑥 <br /> <br />𝑐𝑜𝑠2𝑥 𝑠𝑒𝑛 𝑥 𝑑𝑦𝑑𝑥+𝑐𝑜𝑠3𝑥𝑦=1<br /> <br />
  3. 3. ∫𝑝𝑥=ʃcos𝑥𝑠𝑒𝑛 𝑥𝑑𝑥<br /> µ=senx<br />Multiplicamos la ecuación por el factor integrante:<br />sen x y’+ sen x 𝑐𝑜𝑠𝑥𝑠𝑒𝑛𝑥𝑦=𝑠𝑒𝑛𝑥𝑐𝑜𝑠2𝑠𝑒𝑛𝑥 <br />sen x y’+ cos x y =1𝑐𝑜𝑠2 𝑥<br /> <br />Integramos:<br />
  4. 4. 𝑑(𝑠𝑒𝑛 𝑥 𝑦)𝑑𝑥=𝑠𝑒𝑐2𝑥<br />ʃ 𝑑(𝑠𝑒𝑛 𝑥 𝑦)𝑑𝑥𝑑𝑥=ʃ𝑠𝑒𝑐2𝑥 𝑑𝑥<br />y sen x = tan x + c<br />ysen x =𝑠𝑒𝑛𝑥𝑐𝑜𝑠𝑥+𝑐<br /> <br />Seguimos integrando:<br />
  5. 5. y=𝑠𝑒𝑛 𝑥cos𝑥 𝑠𝑒𝑛 𝑥+𝑐𝑠𝑒𝑛 𝑥<br />y= 1cos𝑥+𝑐𝑠𝑒𝑛 𝑥<br />Finalmente encontramos:<br />y= sec x + csc x<br /> <br />Despejamos y, obtenemos el resultado.<br />

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