The Differential Equation

Definition 3.1 - Ordinary Differential Equation

Let $f(x)$ define a function of $x$ on an interval . By an ordinary differential equation we mean an equation involving $x$, the function $f(x)$ and one or more of its derivatives.

Note. It's customary to replace $f(x)$ by $y$.

Definition 3.2 - Order of a Differential Equation

The order of a differential equation is the order of the highest derivative involved in the equation.

For example, the differential equation $y^{\prime }=e^x$ is of the first order, and the differential equation $y^{″}+y^{\prime }=3y$ is of the second order.

Definition 3.4 - Explicit Solution Let $y=f(x)$ define $y$ as a function of $x$ on an interval . We say that the function $f(x)$ is an explict solution or simply a solution of an ordinary differential equation involving $x$, $f(x)$, and its derivatives, if it satisfies th equation for every $x$ in $I$, i.e., if we replace $y$ by $f(x)$, $y^{\prime }$ by $f^{\prime }(x)$, $y^{″}$ by $f^{″}(x)$, $\cdots$, $y^{(n)}$ by $f^{(n)}(x)$, the differential equation reduces to an identity in $x$. In mathematical symbols, the definition says: the function $f(x)$ is a solution of the differential equation:

$$F(x,y,y^{\prime },\cdots ,y^{(n)}=0$$

if

$$F[x,f(x),f^{\prime }(x),\cdots ,f^{(n)}(x)]=0$$

In more plain english, $f(x)$ is a solution of $F(\dots )=0$ if we replace all of the occurences of $y$ and its derivatives in $F(...)$ with the $f(x)$ and its respective derivatives and the result is a valid equation of only $x$.

To test if $f(x,y)$ is a solution of a differential equation $F(x,y,y^{\prime },\cdots ,y^{(n)})=0$, take the necessary derivatives of $f(x,y)$. Then, in $F$, replace $y$ with $f(x)$, $y^{\prime }$ with $f^{\prime }(x)$, and so on, and see if the result is a valid equation of $x$.

Definition 3.6 - Implicit Solution A relation $f(x,y)=0$ will be called an implicit solution of the differential equation

(3.61) $$F(x,y,y^{\prime },\cdots ,y^{(n)})=0$$

on the interval , if

1. it defines $y$ as an implict function of $x$ on $I$, i.e. if there exists a function $g(x)$ defined on $I$ such that $f[x,g(x)]=0$ for every $x$ in $I$, and if
2. $g(x)$ satisfies 3.61, i.e. if

$$F[x,g(x),g^{\prime }(x),\cdots ,g^{(n)}(x)]=0$$

for every $x$ in $I$.

In more plain english, the relation (not function) $f(x,y)$ is an implicit solution if there is some $g(x)$ that can be chosen to make a function $f(x,g(x))=0$ from $f(x)$. That is, we can chose a branch of $f(x,y)$ to make a function rather than just a relation, and if that function and its derivatives satisfy 3.61 then $f(x,y)$ is considered an implicit solution of the differential equation.

Note that here was have $f(x,y)$ where for an explicit solution we have just $f(x)$.

To test an implicit solution, pick a branch of $f(x,y)$, first pick a branch of $f(x,y)$ that defines a function, and then follow the same procedure as testing an explicit solution. Note that depending on the branch chosen, the resulting function or its derivatives may not be defined on some points in $I$, and so may be solutions for smaller intervals than $I$, or for $I$ excluding some points.