Por favor necesito ayuda con esta ecuación diferencial por factor integrante :Si U = [tex] x ^ { \ alpha } y ^ { \ beta } [ / tex]Encuentre el valor de[tex] \ alpha [ / tex] y de[tex] \ beta [ / ?

Suponiendo que la ecuacion diferencial es : <img src="https://tex.z-dn.net/?f=%20%288y%2B4x%5E2y%5E4%29dx%2B%288x%5E3y%5E3%29dy%3D0%20" />Se multiplica por el factor integrante que ya es proporcionado : <img src="https://tex.z-dn.net/?f=%20%5Bx%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%7D%5D%5B%288y%2B4x%5E2y%5E4%29dx%2B%288x%5E3y%5E3%29dy%5D%3D0%5Bx%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%7D%5D%20" />desarrollamos para encontrar los valores de alpha y beta : <img src="https://tex.z-dn.net/?f=%20%288x%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%2B1%7D%2B4x%5E%7B%5Calpha%2B2%7Dy%5E%7B%5Cbeta%2B4%7D%29dx%2B%288x%5E%7B%5Calpha%2B3%7Dy%5E%7B%5Cbeta%2B3%7D%29dy%3D0%5C%5CSe%5C%20debe%5C%20cumplir%3A%5C%5C%20%5C%5C%5Cfrac%7B%5Cpartial%7B%288x%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%2B1%7D%2B4x%5E%7B%5Calpha%2B2%7Dy%5E%7B%5Cbeta%2B4%7D%29%7D%7D%7B%5Cpartial%7By%7D%7D%3D%5Cfrac%7B%5Cpartial%7B%288x%5E%7B%5Calpha%2B3%7Dy%5E%7B%5Cbeta%2B3%7D%29%7D%7D%7B%5Cpartial%7Bx%7D%7D%20" /><img src="https://tex.z-dn.net/?f=%208%28%5Cbeta%2B1%29x%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%7D%2B4%28%5Cbeta%2B4%29x%5E%7B%5Calpha%2B2%7Dy%5E%7B%5Cbeta%2B3%7D%3D8%28%5Calpha%2B3%29x%5E%7B%5Calpha%2B2%7Dy%5E%7B%5Cbeta%2B3%7D%20" />buscamos la igualdad de los coeficientes : <img src="https://tex.z-dn.net/?f=%20Coeficientes%5C%20de%5C%20%5Cqquad%20x%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%7D%3A%5C%5C8%28%5Cbeta%2B1%29%3D0%20%5Cqquad%20%5Ctherefore%20%5Cqquad%20%5Cbeta%3D-1%5C%5C%20%5C%5CCoeficientes%5C%20de%5C%20%5Cqquad%20x%5E%7B%5Calpha%2B2%7Dy%5E%7B%5Cbeta%2B3%7D%3A%5C%5C4%28%5Cbeta%2B4%29%3D8%28%5Calpha%2B3%29%5C%5C%20%5Cbeta%2B4%3D2%28%5Calpha%2B3%29%5C%5C%20%5Calpha%3D%5Cfrac%7B%5Cbeta-2%7D%7B2%7D%5Cqquad%20%5Ctherefore%20%5Cqquad%20%5Calpha%3D-%5Cfrac%7B3%7D%7B2%7D%20" /><img src="https://tex.z-dn.net/?f=%20U%3Dx%5E%7B%5Calpha%7Dy%5E%7B%5Cbeta%7D%3Dx%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7Dy%5E%7B-1%7D%20" />La ecuacion diferencial entonces sera exacta al multiplicarla por U, obtendremos como resultado de dicha multiplicacion : <img src="https://tex.z-dn.net/?f=%20%288x%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%2B4x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7Dy%5E3%29dx%2B%288x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E2%29dy%3D0%20" />Ahora se procede a resolverla, integramos el factor que multiplica a dx con respecto a x : <img src="https://tex.z-dn.net/?f=%20F%28x%2Cy%29%3D%5Cint%7B%288x%5E%7B-%5Cfrac%7B3%7D%7B2%7D%7D%2B4x%5E%7B%5Cfrac%7B1%7D%7B2%7D%7Dy%5E3%29%20%5C%2Cdx%20%5C%5C%20%5C%5C%3D8%28-2x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%29%2B4%28%5Cfrac%7B2%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%29y%5E3%2Bg%28y%29%5C%5C%20%5C%5C%3D%5Cfrac%7B8%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E3-16x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%2Bg%28y%29%20" />Ahora la derivada parcial con respecto a y del resultado anterior debe ser igual al factor que multiplica a dy en nuestra ecuacion exacta : <img src="https://tex.z-dn.net/?f=%208x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E2%3D%5Cfrac%7B%5Cpartial%7BF%7D%7D%7B%5Cpartial%7By%7D%7D%3D%5Cfrac%7B%5Cpartial%7B%28%5Cfrac%7B8%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E3-16x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%2Bg%28y%29%29%7D%7D%7B%5Cpartial%7By%7D%7D%5C%5C%20%5C%5C8x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E2%3D8x%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E2%2B%5Cfrac%7Bdg%28y%29%7D%7Bdy%7D%5C%5C%20%5C%5C%5Cfrac%7Bdg%28y%29%7D%7Bdy%7D%3D0%20%5Cqquad%20%5Ctherefore%20%5Cqquad%20g%28y%29%3Dconstante%3Dc%20" />La soucion final es entonces : <img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B8%7D%7B3%7Dx%5E%7B%5Cfrac%7B3%7D%7B2%7D%7Dy%5E3-16x%5E%7B-%5Cfrac%7B1%7D%7B2%7D%7D%3Dc%20" />.