Solutions of Differential Equations

A solution of a differential equation is an expression of the dependent variable that satisfies the relation established in the differential equation. For example, the solution of $y'+y-2=4x$ will be some equation y = f(x) such that y and its first derivative, y', satisfy the relation $y'+y-2=4x$. The general solution of a differential equation will have one or more arbitrary constants, depending on the order of the original differential equation (the solution of a first order diff. eq. will have one arbitrary constant, a second order one will have two, etc.).

Examples:

• $y'=2x$. Through simple integration, we can calculate the general solution of this equation to be $y=x^{2}+C$, where C is an arbitrary constant.
• $y'-y=0$. The G.S. is $y=Ce^{x}$. $y'=Ce^{x}$, so $y'-y=0\to Ce^{x}-Ce^{x}=0$, so this solution satisfies the relationship for all arbitrary constants C.
• $y'+y-2=0$. The G.S. is $y=Ce^{-x}+2$. $y'=-Ce^{-x}$, so $0=y'+y-2$ becomes $0=-Ce^{-x}+Ce^{-x}+2-2=0$.