Dasharathi (दाशरथि) is another name for Lord Rama, it means “son of King Dasharatha” (दशरथस्य अपत्यम् पुमान्)
Similarly there is another name called as Daaashrth (दाशरथ), which also means “son of King Dasharatha”
दाशरथि/दाशरथ both names are valid for all the sons of King Dasharatha, including Lakshmana, Bharata and Shatrughana.
Here is an example in Ramayana – King Janaka said that he will his daughter to any son of King Dasharatha, who can lift Shiva Dhanusha.
सुतामयोनिजां सीतां दद्यां दाशरथेरहम् – Ramayana 1.66.26
I will give my daughter Sita, who is not born from a woman, to “any son of King Dasharatha” (दाशरथ)
Dasharathi in Ramayana
गते रामे प्रशान्तात्मा रामो दाशरथिर्धनु:।
वरुणायाप्रमेयाय ददौ हस्ते ससायकम्।।1.77.1।।
When Parashurama departed, son of Dasharatha gave bow and arrow in the hands of Varuna , who has great strength.
श्रुत्वा तज्जामदग्न्यस्य वाक्यं दाशरथिस्तदा।
गौरवाद्यंन्त्रितकथ: पितू राममथाब्रवीत्।।1.76.1।।
son of Dasaratha (Rama) was not saying anything because of respect to his father, but after listening the words of the son of Jamadagni (parashurama), He said to Parashurama:
तत्र श्रियमिवाचिन्त्यां रामो दाशरथिः प्रियाम्।