Como resolver el siguiente te problema con valores iniciales?

<img src="https://tex.z-dn.net/?f=%5Cdfrac%7Bd%5E2x%7D%7Bdt%5E2%7D%2B6%5Cdfrac%7Bdx%7D%7Bdt%7D%2B10x%3D25%5Ccos%204t%5C%5C%20%5C%5C%20%5C%5C%0A%5Cmathcal%7BL%7D%5Cleft%5C%7B%5Cdfrac%7Bd%5E2x%7D%7Bdt%5E2%7D%2B6%5Cdfrac%7Bdx%7D%7Bdt%7D%2B10x%5Cright%5C%7D%3D%5Cmathcal%7BL%7D%5Cleft%5C%7B25%5Ccos%204t%5Cright%5C%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cmathcal%7BL%7D%5Cleft%5C%7B%5Cdfrac%7Bd%5E2x%7D%7Bdt%5E2%7D%5Cright%5C%7D%2B6%5Cmathcal%7BL%7D%5Cleft%5C%7B%5Cdfrac%7Bdx%7D%7Bdt%7D%5Cright%5C%7D%2B10%5Cmathcal%7BL%7D%5Cleft%5C%7Bx%5Cright%5C%7D%3D25%5Cmathcal%7BL%7D%5Cleft%5C%7B%5Ccos%204t%5Cright%5C%7D%0A" />

<img src="https://tex.z-dn.net/?f=%5Cleft%5Bs%5E2%5Cmathcal%7BL%7D%5Cleft%5C%7Bx%5Cright%5C%7D-sx%280%5E-%29-x%27%280%5E-%29%5Cright%5D%2B6%5Cleft%5Bs%5Cmathcal%7BL%7D%5Cleft%5C%7Bx%5Cright%5C%7D-x%280%5E-%29%5Cright%5D%2B10%5Cmathcal%7BL%7D%5Cleft%5C%7Bx%5Cright%5C%7D%3D%5Cdfrac%7B25s%7D%7Bs%5E2%2B4%5E2%7D%5C%5C%20%5C%5C%20%5C%5C%0A%28s%5E2%2B6s%2B10%29%5Cmathcal%7BL%7D%5C%7Bx%5C%7D-%5Cdfrac%7Bs%7D%7B2%7D-3%3D%5Cdfrac%7B25s%7D%7Bs%5E2%2B16%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cmathcal%7BL%7D%5C%7Bx%5C%7D%3D%5Cdfrac%7B%5Cdfrac%7B25s%7D%7Bs%5E2%2B16%7D%2B%5Cdfrac%7Bs%7D%7B2%7D%2B3%7D%7Bs%5E2%2B6s%2B10%7D" />

<img src="https://tex.z-dn.net/?f=%5Cmathcal%7BL%7D%5C%7Bx%5C%7D%3D%5Cdfrac%7B%5Cfrac%7B38%7D%7B51%7Ds%7D%7Bs%5E2%2B6s%2B10%7D%2B%5Cdfrac%7B%5Cfrac%7B28%7D%7B51%7D%7D%7Bs%5E2%2B6s%2B10%7D-%5Cdfrac%7B%5Cfrac%7B25%7D%7B102%7Ds%7D%7Bs%5E2%2B16%7D%2B%5Cdfrac%7B%5Cfrac%7B200%7D%7B51%7D%7D%7Bs%5E2%2B16%7D%5C%5C%20%5C%5C%20%5C%5C%0Ax%28t%29%3D%5Cdfrac%7B38%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7Bs%7D%7Bs%5E2%2B6s%2B10%7D%5Cright%5C%7D%2B%5Cdfrac%7B28%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7B1%7D%7Bs%5E2%2B6s%2B10%7D%5Cright%5C%7D-%5Ccdots%5C%5C%20%5C%5C%0A-%5Cdfrac%7B25%7D%7B102%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7Bs%7D%7Bs%5E2%2B16%7D%5Cright%5C%7D%2B%5Cdfrac%7B200%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7B1%7D%7Bs%5E2%2B16%7D%5Cright%5C%7D%5C%5C%20%5C%5C%20%5C%5C%0A" />

<img src="https://tex.z-dn.net/?f=x%28t%29%3D%5Cdfrac%7B38%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7Bs%7D%7B%28s%2B3%29%5E2%2B1%7D%5Cright%5C%7D%2B%5Cdfrac%7B28%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7B1%7D%7B%28s%2B3%29%5E2%2B1%7D%5Cright%5C%7D-%5Ccdots%5C%5C%20%5C%5C%0A-%5Cdfrac%7B25%7D%7B102%7D%5Ccos%284t%29%2B%5Cdfrac%7B50%7D%7B51%7D%5Csin%284t%29" />

<img src="https://tex.z-dn.net/?f=x%28t%29%3D%5Cdfrac%7B38%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7Bs%2B3%7D%7B%28s%2B3%29%5E2%2B1%7D%5Cright%5C%7D-%5Cdfrac%7B86%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7B1%7D%7B%28s%2B3%29%5E2%2B1%7D%5Cright%5C%7D-%5Ccdots%5C%5C%20%5C%5C%0A-%5Cdfrac%7B25%7D%7B102%7D%5Ccos%284t%29%2B%5Cdfrac%7B50%7D%7B51%7D%5Csin%284t%29%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BRecordando%20la%20propiedad%20de%20traslaci%5C%27on%7D%5C%5C%20%5C%5C%0A%5Cboxed%7B%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7BF%28s-%5Comega%29%5Cright%5C%7D%3De%5E%7B%5Comega%20t%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7BF%28s%29%5Cright%5C%7D%7D" />

<img src="https://tex.z-dn.net/?f=x%28t%29%3D%5Cdfrac%7B38e%5E%7B-3t%7D%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7Bs%7D%7Bs%5E2%2B1%7D%5Cright%5C%7D-%5Cdfrac%7B86e%5E%7B-3t%7D%7D%7B51%7D%5Cmathcal%7BL%7D%5E%7B-1%7D%5Cleft%5C%7B%5Cdfrac%7B1%7D%7Bs%5E2%2B1%7D%5Cright%5C%7D-%5Ccdots%5C%5C%5C%5C%0A%5Ccdots-%5Cdfrac%7B25%7D%7B102%7D%5Ccos%284t%29%2B%5Cdfrac%7B50%7D%7B51%7D%5Csin%284t%29%5C%5C%20%5C%5C%20%5C%5C%20%5C%5C%0A%5Cboxed%7Bx%28t%29%3D%5Cdfrac%7B38e%5E%7B-3t%7D%7D%7B51%7D%5Ccos%20t-%5Cdfrac%7B86e%5E%7B-3t%7D%7D%7B51%7D%5Csin%20t-%5Cdfrac%7B25%7D%7B102%7D%5Ccos%284t%29%2B%5Cdfrac%7B50%7D%7B51%7D%5Csin%284t%29%7D" />.