There are 2 answers:

1. 1806

at his 43 years age the year is 1849

so, his age = 43 in year 43^2=1849

2. 1892

at his 44 years age the year is 1936

so,his age = 44 in year 44^2=1936

so, if rohan is born and died in 19th century then,he was born in 1806

but the question is asked Rohan was born in 19th century, he will be x year old in x^2 year. Find in which year was he born?

so, 1806 or 1892