Using DSolve to Find  Symbolic Solutions to Differential Equations: General Homogeneous Case

Mathematica can solve the the general homogeneous first-order linear ODE:



The dummy integration variables  (K[1] in the above) and any integration constants  (C[1] above) are picked by Mathematica .  Note that we asked Mathematica to solve the most general form of homogeneous linear first-order ODE, and we got a solution in a very general form that is equivalent to that found in textbooks.  Here is a more specific problem and the solution found by Mathematica:



There is an integration constant above, that will take on a specific value if an additional condition (such as an initial condition, or a boundary condition) is specified. Below, we stipulate that y[0] is 4; so we should get the result with C[1] specified; y[0] = C[1] = 4



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