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Ordinary Differential Equations

Let \(y(x)\) denote a function of \(x\). A differential equation is a function that captures the relations between the derivatives of \(y\) and the functions of \(x\). An ordinary differential equation will contain \(y\) as only the function of \(x\) (independent variable). An equation containing the partial derivatives will be call a partial differential equation.

\[ \begin{aligned} F(x, y^{(1)}, y^{(2)}, \ldots, y^{(n)}) = 0 \end{aligned} \]

where \(y^{(1)}, y^{(2)}, \ldots, y^{(n)}\) denote the various derivates of \(y\).

The order of a differential equation is the highest derivative in the equation. For the above case, the order will be \(n\) assuming that a term containing \(y^{(n)}\) exists in the equation with non-zero coefficient.

Similarly, the degree of a differential equation is the power to which the highest derivative is raised in the equation. Suppose a term \((y^{(n)})^{m}\) existed in the equation where \(n\) is the order, then the degree will be \(m\).

A differential equation is linear if it can be expressed as follows

\[ \begin{aligned} L(y(x)) &= L(y^{(1)}, y^{(2)}, \ldots, y^{(n)}) = f(x)\newline a_{0}(x)y^{(n)}+ a_{1}(x)y^{(n-1)} + \cdots + a_{n}(x)y &= f(x) \end{aligned} \]

where \(L\) is a linear transformation from the space of functions that are derivatives, to the space of functions. The coefficients in this linear combination can themselves be functions of \(x\), with \(a_{0}(x) \neq 0\). Any equation not expressed in this form will be termed non-linear.

A simple example of DE in real life is the motion of a pendumlum modeled as

\[ \begin{aligned} \frac{d^{2}\theta}{dt^{2}} + \frac{g}{l}\sin\theta \end{aligned} \]

Particular solution

Suppose we have a differential equation of the form \(y^{(1)} = f(x,y)\) and the solution to this equation is \(y = g(x) + c\), where \(c\) is an arbitrary constant. Then, this solution is called a general solution since it represents a family of curves. The solution can also be of the form \(y = cg(x)\), or anything else depending on the differential equation itself.

If we choose a specific value of the constant, we get a particular solution to the problem. The constant can be determined using an initial condition like \(y(x_{0}) = y_{0}\). An ODE together with the initial condition is called an initial value problem.

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