The most determined character in The Railway children would be Bobbie. This was shown when she still had hopes that her father who was accused of being an antagonist was still innocent. she tried her utmost best to do anything to free her father. She even tried to convince the old gentleman that her father was a good man and she succeeded.Therefore, bobbie is the most determined. 
Bobbie. why? determined means, to act immediately or to have an idea to accomplish a problem. therefore, bobbie suggested to write to the old gentleman to ask for help because their mother is sick. they dont have enough money to buy the items the doctor had listed. another way is, when there was a landslide at the railway. the children immediately ran to stop the oncoming 11.45(i think,i forgot the name of the train. hehe) train. when the train is coming near, she cut off the red petticoat from her skirt and used it as a flag. this helped from stopping the train from crashing into the landslide. lastly, ----