1. Resolver la siguiente ecuación diferencial por el método de variación de parámetros : y´´ + y = sec2x 2. Resolver la siguiente ecuación diferencial por el método de coeficientes indeterminados : y´´ + 3y´ + 2y = 3x + 1 3. Encontrar el operador diferencial que anule a : a. X + 3xye6x b. (x3 - 2x)(x2 – 1) c. Xex.
ax² + bx + c = 0
En resumen
1) resolvamos la EDO homogénea <img src="https://tex.z-dn.net/?f=y%27%27%2By%3D0%5C%5C%20%0Ar%5E2%2B1%3D0%5Ciff%20r%3D%5Cpm%20i%5C%5C%20%5C%5C%0Ay_h%3DC_1%20%5Ccos%20x%2BC_2%5Csin%20x" /> para resolver la ecuación<img src="https://tex.z-dn.net/?
1) resolvamos la EDO homogénea
<img src="https://tex.z-dn.net/?f=y%27%27%2By%3D0%5C%5C%20%0Ar%5E2%2B1%3D0%5Ciff%20r%3D%5Cpm%20i%5C%5C%20%5C%5C%0Ay_h%3DC_1%20%5Ccos%20x%2BC_2%5Csin%20x" />
para resolver la ecuación<img src="https://tex.z-dn.net/?f=y%27%27%2By%3D%5Csec%202x" /> utilizamos la matriz Wronskiana
<img src="https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A%5Ccos%20x%26%5Csin%20x%5C%5C%0A%28%5Ccos%20x%29%27%26%28%5Csin%20x%29%27%0A%5Cend%7Bmatrix%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0Au_1%27%5C%5Cu_2%27%0A%5Cend%7Bmatrix%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A0%5C%5C%5Csec%202x%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A%5Ccos%20x%26%5Csin%20x%5C%5C%0A-%5Csin%20x%26%5Ccos%20x%0A%5Cend%7Bmatrix%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0Au_1%27%5C%5Cu_2%27%0A%5Cend%7Bmatrix%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A0%5C%5C%5Csec%202x%0A%5Cend%7Bmatrix%7D%5Cright%5D" />
Como la matriz ortonormal entonces
<img src="https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Bmatrix%7D%0Au_1%27%5C%5Cu_2%27%0A%5Cend%7Bmatrix%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A%5Ccos%20x%26-%5Csin%20x%5C%5C%0A%5Csin%20x%26%5Ccos%20x%0A%5Cend%7Bmatrix%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Bmatrix%7D%0A0%5C%5C%5Csec%202x%0A%5Cend%7Bmatrix%7D%5Cright%5D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cdisplaystyle%0Au%27_1%28x%29%3D-%5Csin%20x%5Csec%202x%5Cto%20u_1%28x%29%3D%5Cint%20%5Cdfrac%7B-%5Csin%20x%7D%7B2%5Ccos%5E2%20x-1%7Ddx%20%5C%5C%20%5C%5C%0Au_1%28x%29%3D%20%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%5Cln%20%5Cleft%28%5Cdfrac%7B%5Csqrt%7B2%7D%5Ccos%20x-1%7D%7B%5Csqrt%7B2%7D%5Ccos%20x%2B1%7D%5Cright%29%5C%5C%20%5C%5C%0Au%27_2%28x%29%3D%5Ccos%20x%20%5Csec%202x%5C%5C%20%5C%5C%0Au_2%28x%29%3D%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%5Cln%20%5Cleft%28%5Cdfrac%7B%5Csqrt%7B2%7D%5Csin%20x%2B1%7D%7B%5Csqrt%7B2%7D%5Csin%20x-1%7D%5Cright%29%5C%5C%20%5C%5C" />
Por ende
<img src="https://tex.z-dn.net/?f=y%3D%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%5Cln%20%5Cleft%28%5Cdfrac%7B%5Csqrt%7B2%7D%5Ccos%20x-1%7D%7B%5Csqrt%7B2%7D%5Ccos%20x%2B1%7D%5Cright%29%5Ccos%20x%2B%5Cdfrac%7B%5Csqrt%7B2%7D%7D%7B4%7D%5Cln%20%5Cleft%28%5Cdfrac%7B%5Csqrt%7B2%7D%5Csin%20x%2B1%7D%7B%5Csqrt%7B2%7D%5Csin%20x-1%7D%5Cright%29%5Csin%20x%2B%5Ccdots%5C%5C%5C%5C%0A%5Chspace%2A%7B7cm%7D%2BC_1%5Ccos%20x%20%2BC_2%5Csin%20x" />.