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.
• $y''+y'-2y=0$. The G.S. is $y=Ce^{x}+De^{-2x}$. $y'=Ce^{x}-2De^{-2x}$ and $y''=Ce^{x}+4De^{-2x}$, so $0=y''+y'-2y$ becomes $0=Ce^{x}+4De^{-2x}+Ce^{x}-2De^{-2x}-2(Ce^{x}+De^{-2x})=Ce^{x}+Ce^{x}-2Ce^{x}+4De^{-2x}-2De^{-2x}-2De^{-2x})=0$.

The particular solution of a differential equation can be solved if we have enough points to solve for the arbitrary constants (see initial value problems for more).

Existence of solutions

Solving differential equations is not like solving algebraic equations. Not only are their solutions often unclear, but whether solutions are unique or exist at all are also notable subjects of interest.

For first order initial value problems, the Peano existence theorem gives one set of circumstances in which a solution exists. Given any point $(a,b)$ in the xy-plane, define some rectangular region $Z$, such that $Z=[l,m]\times [n,p]$ and $(a,b)$ is in the interior of $Z$. If we are given a differential equation ${\frac {dy}{dx}}=g(x,y)$ and the condition that $y=b$ when $x=a$, then there is locally a solution to this problem if $g(x,y)$ and ${\frac {\partial g}{\partial x}}$ are both continuous on $Z$. This solution exists on some interval with its center at $a$. The solution may not be unique.

However, this only helps us with first order initial value problems. Suppose we had a linear initial value problem of the nth order:

$f_{n}(x){\frac {d^{n}y}{dx^{n}}}+\cdots +f_{1}(x){\frac {dy}{dx}}+f_{0}(x)y=g(x)$

such that

$y(x_{0})=y_{0},y'(x_{0})=y'_{0},y''(x_{0})=y''_{0},\ldots$

For any nonzero $f_{n}(x)$, if $\{f_{0},f_{1},\ldots \}$ and $g$ are continuous on some interval containing $x_{0}$, $y$ is unique and exists.

Resources

• Differential Equations, University of Glascow
• General and Particular Solutions, Simply Math

Licensing

Content obtained and/or adapted from:

• Differential equation, Wikipedia under a CC BY-SA license