<p>A <strong>differential equation</strong> is one that relates the <em>independent variable</em> \(x\), the <em>dependent variable</em> \(y\) as a function of \(x\) and its derivative with respect to the independent variable \(x\). </p>
<p>Examples of differential equations are:</p>
<ol class="i-list"><li><div class="list-item-paragraph">\[
\frac{\mathrm{d}y}{\mathrm{d}x} = \cos x.
\]</div></li></ol>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">\[
x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = y^2 + 2.
\]</div></li></ol>
<ol class="i-list" start="3"><li><div class="list-item-paragraph">\[
\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 6x - 2.
\]</div></li></ol>
<p>A <strong>first order differential equation</strong> involves only the first derivative of the dependent variable with respect to the independent one. <br>Examples (i) and (ii) above are examples of first order differential equations.</p>
<p>A <strong>solution of a differential equation</strong> is a function \(y(x)\) -- expressing an explicit relation (through an equation) between the independent and dependent variables -- which satisfies the differential equation for all values of \(x\) in the domain.</p>
<p>In this topic, we will be exploring different methods of solving first order differential equations.</p>

A **differential equation** is one that relates the *independent variable* \(x\), the *dependent variable* \(y\) as a function of \(x\) and its derivative with respect to the independent variable \(x\). 

\[\frac{\mathrm{d}y}{\mathrm{d}x} = \cos x\]

\[
\frac{\mathrm{d}y}{\mathrm{d}x} = \cos x.
\]

\[x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = y^2 + \(2\)\]

\[
x^2 \frac{\mathrm{d}y}{\mathrm{d}x} = y^2 + 2.
\]

\[\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \(6\)x - \(2\)\]

\[
\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = 6x - 2.
\]

A **first order differential equation** involves only the first derivative of the dependent variable with respect to the independent one. 
Examples (i) and (ii) above are examples of first order differential equations.

A **solution of a differential equation** is a function \(y(x)\) -- expressing an explicit relation (through an equation) between the independent and dependent variables -- which satisfies the differential equation for all values of \(x\) in the domain.

In this topic, we will be exploring different methods of solving first order differential equations.

Definition of a differential equation

Definition of a differential equation