Which you are asked to use (e.g., Variable Separable, Exact, etc.) If you need a step-by-step derivation of a specific theorem
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, often achieved by eliminating arbitrary constants from a given relation between variables. 2. Definitions and Classification Ordinary Differential Equations (ODEs): Involve functions of only one independent variable. Order and Degree:
Let [ y^\prime+p(x),y = 0,\qquad p\in C^0(I) ] be a first‑order linear homogeneous ODE on an interval (I\subset\mathbbR). Then any non‑zero solution (y_1(x)) never vanishes on (I) and the set (y_1) is a fundamental set of solutions . Consequently every solution can be written as [ y(x)=C,y_1(x),\quad C\in\mathbbR. ]