To say that you have found the general homogeneous solution means that this function solves the homogeneous equation for every choice of the constant $C_1$ and every solution of the homogeneous equation is of this form for some choice of $C_1$. You can actually have more than one particular solution to a DEQ.
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For the adjacent town, see Mukwonago (town), Wisconsin. Mukwonago is a village in Wisconsin. The population was 7,355 at the 2010 census. The village is located mostly within the Town of Mukwonago in Waukesha County, with a small portion extending into the Town of East Troy in Walworth County. Of its population, 7,254 were in Waukesha County, and 101 were in Walworth County.