Section 4.1 Solution of a Differential Equation
Let the differential equation
\begin{equation}
\frac{\,dy}{\,dx}=\frac{1}{x}\tag{4.1.1}
\end{equation}
Integrating both sides of eqn. (4.1.1), we get -
\begin{equation}
y=\int\frac{1}{x}\,dx = \log x+C\tag{4.1.2}
\end{equation}
The eqn. (4.1.2) is called the general solution of differential eqn. (4.1.1) and \(C\) is an arbitrary constant. Consider the another differential equation
\begin{equation}
\frac{\,d^{2}y}{\,dx^{2}}=x\tag{4.1.3}
\end{equation}
Integrating both sides, we get -
\begin{equation*}
\int\,d\left(\frac{\,dy}{\,dx}\right)=\int x\,dx
\end{equation*}
or,
\begin{equation}
\frac{\,dy}{\,dx}=\frac{x^{2}}{2}+C \tag{4.1.4}
\end{equation}
Integrating again, we get -
\begin{equation}
y=\frac{x^{3}}{6}+Cx+D\tag{4.1.5}
\end{equation}
The eqn. (4.1.5) is the general solution of differential eqn. (4.1.3) and C and D are two arbitrary constants. Hence it is clear from eqns. (4.1.2) and (4.1.5) that if the order of differential equation is one the solution contains only one arbitrary constant but if it is second ordered then its solution also contains two arbitrary constants. The number of arbitrary constants in the solution is equal to the order of the differential equation. An equation containing dependent variable y, independent variable x and free from derivative which also satisfies the differential equation is called the solution of the differential equation. The general solution becomes a particular solution by assigning the particular value of arbitrary constant. Suppose we find C = 1 and D = 2 for eqn. (4.1.5) then, \(y=\frac{x^{3}}{6}+Cx+D\) is the particular solution of differential eqn. (4.1.3).
