I searched all the churches in the city of Aarhus but could not find the birth of Gerda. However I did find the birth of John Ivar Svensson on 30 Mar 1910 in den kgl Fødselsskiftelse, Copenhagen. The mother is unmarried and unnamed and the father is not mentioned. Den kgl. Fødselsskiftelse was a hospital where a woman could give birth without giving any information. I found Hilda Svendson and John Ivar Svendsen in the 1911 Copenhagen census.
I have saved the two documents and will send my e-mail address privately if you want them.
Here is Hilda in the police register that was kept on all residents of Copenhagen. http://www.politietsregisterblade.dk/index.php?option=com_sf...
I hope this helps