Why was Dasaratha so emotionally attached only to Rama?


Lord Rama in Ramcharitmanas Uttar kanda answers this question giving an analogy to that as:

एक पिता के बिपुल कुमारा ।
होहिं पृथक गुन सील अचारा ।।
कोउ पंडित कोउ तापस ग्याता ।
कोउ धनवंत सूर कोउ दाता।।
कोउ सर्वग्य धर्मरत कोइ।
सब पर पितहि प्रीति सम होई।।
कोउ पितु भगत बचन मन कर्मा ।
सपनेहुँ जान न दूसर धर्मा ।।
सो सुत प्रिय पितु प्रान समाना ।
जद्यपि सो सब भाॅति अयाना।।
एही बिधि जीव चराचर जेते ।
त्रिजग देव नर असुर समेते।।
अखिल बिस्व यह मोर उपाया ।
सब पर मोहि बराबरि दाया।।
तिन्ह मह जो परिहरि मद माया ।
भजै मोहि मन बच अरु काया।।

पुरुष नपुंसक नारि वा जीव चराचर कोइ ।
सर्व भाव भज कपट तजि मोहिपरम प्रिय सोइ ।! ८७
सत्य कहउँ खग तोहि सुचि सेवक मम प्रान प्रिय ।
अस बिचारी भजु मोहि परिहरिआस भरोस सब । ।

This translates as

A father has a number of sons, each differing from the others in character, temper and conduct. One is learned, another given to austerities, a third spiritually enlightened, a fourth rich, a fifth possessed of valour, a sixth charitably disposed, a seventh all-wise and an eighth intent on piety; but the father equally loves all. A ninth son is devoted to his father in thought, > word and deed and never dreams of any other duty. This is the son whom the father loves as his own life, though he be a perfect ignoramus. In a like manner all animate and inanimate beings, including the > subhuman species, gods, men and demons in short, the whole of this universe is My creation and I am equally compassionate to all. Of these, however, he > who adores Me in thought, word and deed, forswearing arrogance and wiles.

Be it man, woman or one lacking the characteristics of both, or, for the matter of that, any living being whatsoever of the animate or inanimate world, he who adores Me with all his being, giving up all guile, is supremely dear to Me. O bird, I tell you in all sincerity that a guileless servant is dear to Me as life. Realizing this worship Me,
abandoning all other hope and reliance.

Hence, although all the four brothers were devoted to father but devotion of was Rama was incomparable to other brothers. Hence Rama was Prana-Priya to Dasharatha.

Listen it here from 1:41:30

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