Hallar la solución general de la siguiente ecuación como una serie de potencial alrededor del punto x = 02y" + xy´ + y = 0?
Hallar la solución general de la siguiente ecuación como una serie de potencial alrededor del punto x = 0 2y" + xy´ + y = 0.
Hallar la solución general de la siguiente ecuación como una serie de potencial alrededor del punto x = 0 2y" + xy´ + y = 0.
ax² + bx + c = 0
En resumen
<img src="https://tex.z-dn.net/?
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<img src="https://tex.z-dn.net/?f=%5Ctext%7BSustituimos%7D%3A%5C%5C%5C%5C%0A%5Cdisplaystyle%0A2%5Csum_%7Bn%3D2%7D%5E%7B%5Cinfty%7Da_nn%28n-1%29x%5E%7Bn-2%7D%2Bx%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_nnx%5E%7Bn-1%7D%2B%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7Da_nx%5En%3D0%5C%5C%5C%5C%5C%5C%0A%5Csum_%7Bn%3D2%7D%5E%7B%5Cinfty%7D2a_nn%28n-1%29x%5E%7Bn-2%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_nnx%5E%7Bn%7D%2B%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7Da_nx%5En%3D0%5C%5C%5C%5C%5C%5C%0A%5Ctext%7BIgualando%20potencias%7D%3A%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D2a_%7Bn%2B2%7D%28n%2B2%29%28n%2B1%29x%5E%7Bn%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_nnx%5E%7Bn%7D%2B%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7Da_nx%5En%3D0%5C%5C%5C%5C%5C%5C%0A%5Ctext%7BIgualando%20sub%20%5C%27indices%7D%5C%5C%20%5C%5C%0A4a_2%2Ba_0%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D2a_%7Bn%2B2%7D%28n%2B2%29%28n%2B1%29x%5E%7Bn%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_nnx%5E%7Bn%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_nx%5En%3D0%5C%5C%5C%5C%5C%5C%0A4a_2%2Ba_0%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cleft%5B2a_%7Bn%2B2%7D%28n%2B2%29%28n%2B1%29%2Ba_n%28n%2B1%29%5Cright%5Dx%5En%3D0%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Ctext%7BComparando%20coeficientes%3A%7D%5C%5C%5C%5C%0A4a_2%2Ba_0%3D0%5Cto%20a_2%3D-%5Cdfrac%7Ba_0%7D%7B4%7D%5C%5C%5C%5C%0A2a_%7Bn%2B2%7D%28n%2B2%29%28n%2B1%29%2Ba_n%28n%2B1%29%3D0%5C%3B%2C%5C%3B%20%5Ctext%7Bpara%20%7Dn%5Cgeq1%5C%5C%5C%5C%0Aa_%7Bn%2B2%7D%3D-%5Cdfrac%7Ba_n%7D%7B2%28n%2B2%29%7D%5C%3B%2C%5C%3Bn%5Cgeq1%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Ctext%7BProbemos%3A%7D%5C%5C%20%5C%5C%0A%5Ctext%7BCon%20%7Dn%3D1%3A%5C%3B%5C%3B%20a_3%3D-%5Cdfrac%7Ba_1%7D%7B2%283%29%7D%5C%5C%5C%5C%0A%5Ctext%7BCon%20%7Dn%3D2%3A%5C%3B%5C%3B%20a_4%3D-%5Cdfrac%7Ba_2%7D%7B2%284%29%7D%3D%5Cdfrac%7Ba_0%7D%7B2%5E3%284%29%7D%5C%5C%5C%5C%0A%5Ctext%7BCon%20%7Dn%3D3%3A%5C%3B%5C%3B%20a_5%3D-%5Cdfrac%7Ba_3%7D%7B2%285%29%7D%3D-%5Cdfrac%7Ba_1%7D%7B2%5E2%283%5Ccdot%205%29%7D%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Ctext%7BCon%20%7Dn%3D4%3A%5C%3B%5C%3B%20a_6%3D-%5Cdfrac%7Ba_4%7D%7B2%286%29%7D%3D-%5Cdfrac%7Ba_0%7D%7B2%5E4%284%5Ccdot6%29%7D%5C%5C%5C%5C%0A%5Ctext%7BCon%20%7Dn%3D5%3A%5C%3B%5C%3B%20a_7%3D-%5Cdfrac%7Ba_5%7D%7B2%287%29%7D%3D%5Cdfrac%7Ba_1%7D%7B2%5E3%283%5Ccdot5%5Ccdot7%29%7D%5C%5C%5C%5C%0A%5Cvdots%5C%5C%20%5C%5C%0Aa_%7B2n-1%7D%3D%5Cdfrac%7B%28-1%29%5E%7Bn%2B1%7Da_1%7D%7B2%5E%7Bn%7D%281%5Ccdot3%5Ccdot5%5Ccdot7%5Ccdots%282n%2B1%29%29%7D%5C%5C%5C%5C%0Aa_%7B2n-1%7D%3D%5Cdfrac%7B%28-1%29%5E%7Bn%2B1%7Da_1%282%5Ccdot4%5Ccdot6%5Ccdots%282n%29%29%7D%7B2%5E%7Bn%7D%282n%2B1%29%21%7D%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0A%5Cboxed%7Ba_%7B2n-1%7D%3D%5Cdfrac%7B%28-1%29%5E%7Bn%2B1%7Dn%21%7D%7B%282n%2B1%29%21%7Da_1%5C%3B%2C%5C%3Bn%5Cgeq%201%7D%5C%5C%5C%5C%5C%5C%0Aa_%7B2n%7D%3D%5Cdfrac%7B%28-1%29%5En%7D%7B2%5En%282%5Ccdot4%5Ccdot6%5Ccdots%282n%29%29%7Da_0%5C%5C%5C%5C%0A%5Cboxed%7Ba_%7B2n%7D%3D%5Cdfrac%7B%28-1%29%5E%7Bn%7D%7D%7B4%5E%7Bn%7Dn%21%7Da_0%5C%3B%2C%5C%3B%20n%5Cgeq1%7D%5C%5C%5C%5C%0Ay%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7Da_nx%5En%5C%5C%5C%5C%0Ay%3D%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7Da_%7B2n%7Dx%5E%7B2n%7D%2B%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7Da_%7B2n-1%7Dx%5E%7B2n-1%7D%5C%5C%5C%5C%0A%5Cboxed%7By%3Da_0%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%5Cdfrac%7B%28-1%29%5E%7Bn%7D%7D%7B4%5E%7Bn%7Dn%21%7Dx%5E%7B2n%7D%2Ba_1%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cdfrac%7B%28-1%29%5E%7Bn%2B1%7Dn%21%7D%7B%282n%2B1%29%21%7Dx%5E%7B2n-1%7D%7D%0A" />.
El primer paso es eliminar los símbolos de valor absoluto, lo cual lleva a dos igualdades, ya que la función devuelve el mismo valor cuando la cantidad encerrada entre las barras es positiva y cuando es negativa ; es…
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