Section 16.2 Finding the Particular Solution given Initial Conditions
In Section 16.1, we saw that solving a differential equation may result in one or more arbitrary constants in the solution. This general solution is essentially an entire family of functions that are collectively all solutions to the differential equation. However, if we are given additional information about the function in the form of an initial condition, we may be able to give a more specific function as a solution.
These initial conditions are typically written in the form \(y(x_0)=y_0\) for constants \(x_0\) and \(y_0\text{.}\) This means that when \(x=x_0\text{,}\) we must have \(y=y_0\) and the solution is a curve that passes through the point \((x_0,y_0)\text{.}\) The combination of a differential equation with one or more initial conditions is known as an initial value problem and the solution is known as a particular solution.
Continuing the example from Section 16.1, we saw that the differential equation
\begin{equation*}
y'=x^2y
\end{equation*}
has the general solution \(y(x) = A\,{{ e}^{1/3\,{x}^{3}}}\text{.}\) Now suppose that we add the initial condition \(y(0)=5\) so that the function \(y(x)\) goes through the point \((0,5)\text{.}\) In this case, we have the initial value problem
\begin{equation*}
\left\lbrace \begin{array}{l}
y'= x^2y \\
y(0) = 5
\end{array}\right.\text{.}
\end{equation*}
To include this initial condition when solving the differential equation in Maple, we add it into the
dsolve() command. The initial value problem (differential equation and conditions) must be input within square brackets to create a list.
> de1 := y'(x)= x^2*y(x);
\begin{equation*}
\displaystyle de\mathit{1}\, := \,{\frac {d}{dx}}y \left( x \right) ={x}^{2}y \left( x \right)
\end{equation*}
> dsolve( [de1, y(0) = 5], y(x));
\begin{equation*}
\displaystyle y \left( x \right) =5\,{{ e}^{1/3\,{x}^{3}}}
\end{equation*}
The particular solution to \(y'=x^2y\) with the initial condition \(y(0)=5\) is \(y \left( x \right) =5\,{{ e}^{1/3\,{x}^{3}}}\text{.}\)
