FIRST ORDER DIFFERENTIAL EQUATION

Introduction:

\[\hspace{3cm}\ Since\ the\ time\ of\ Newton,\ physical\ problems\ have\ been\]\[investigated\ by\ formulating\ them\ mathematically\ as\ differential\ equations.\]\[Many\ mathematical\ models\ in\ engineering\ employ\ differential\ equations.\ extensively.\]
\[\hspace{3cm}\ In\ the\ first\ order\ differential\ equation,\ say \frac{dy}{dx}\ =\ f (x, y),\]\[it\ is\ sometimes\ possible\ to\ group\ function\ of\ x\ with\ dx\ on\ one\ side\]\[and\ function\ of\ y\ with\ dy\ on\ the\ other\ side.\]\[This\ type\ of\ equation\ is\ called\ variables\ separable\ differential\ equations.\]\[The\ solution\ can\ be\ obtained\ by\ integrating\ both\ sides\ after\ separating\ the\ variables.\]

Example  1:

\[1.\ \color{red}{Solve:\ xdy\ +\ ydx\ =\ 0}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[xdy\ =\ -\ ydx\]
\[\frac{dy}{y}\ =\ -\ \frac{dx}{x}\]
\[Integrating\ on\ both\ sides\]
\[\int \frac{dy}{y}\ =\ -\ \int \frac{dx}{x}\]
\[log\ y\ =\ -\ log\ x\ +\ log\ c\]
\[log\ y\ +\ log\ x\ =\ log\ c\]
\[\boxed{xy\ =\ c}\]

Example  2:

\[2.\ \color{red}{Solve:\ xdx\ +\ ydy\ =\ 0}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[xdx\ =\ -\ ydy\]
\[Integrating\ on\ both\ sides\]
\[\int x\ dx\ =\ -\ \int y\ dy \]
\[\boxed{\frac{x^2}{2}\ =\ -\frac{y^2}{2}\ +\ c}\]

Example  3:

\[3.\ \color{red}{Solve:\ \frac{dy}{dx}\ =\ y\ e^x}\ \hspace{18cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[\frac{dy}{dx}\ =\ y\ e^x\]
\[\frac{dy}{y}\ =\ e^x\ dx\]
\[Integrating\ on\ both\ sides\]
\[\int \frac{dy}{y}\ =\ \int e^x\ dx\]
\[\boxed{log\ y\ =\ e^x\ +\ c}\]

Example  4:

\[4.\ \color{red}{Solve:\ \frac{dy}{dx}\ =\ \frac{1\ +\ y^2}{1\ +\ x^2}}\ \hspace{18cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[\frac{dy}{dx}\ =\ \frac{1\ +\ y^2}{1\ +\ x^2}\]
\[(1\ +\ x^2)dy\ =\ (1\ +\ y^2)dx\]
\[\frac{dy}{1\ +\ y^2}\ =\ \frac{dx}{1\ +\ x^2}\]
\[Integrating\ on\ both\ sides\]
\[\int \frac{dy}{1\ +\ y^2}\ =\ \int \frac{dx}{1\ +\ x^2}\]
\[\boxed{tan^{-1}\ y\ =\ tan^{-1}\ x\ +\ c}\]

Example  5:

\[5.\ \color{red}{Solve:\ \frac{dy}{dx}\ +\ \frac{1\ +\ x^2}{1\ +\ y^2}\ =\ 0}\ \hspace{18cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[\frac{dy}{dx}\ =\ -\ \frac{1\ +\ x^2}{1\ +\ y^2}\]
\[(1\ +\ y^2)dy\ =\ -\ (1\ +\ x^2)dx\]
\[Integrating\ on\ both\ sides\]
\[\int (1\ +\ y^2)dy\ =\ -\ \int (1\ +\ x^2)dx\]
\[\boxed{y\ +\ \frac{y^3}{3}\ =\ -\ x\ -\ \frac{x^3}{3}\ +\ c}\]

Example  6:

\[6.\ \color{red}{Solve:\ \frac{dy}{dx}\ =\ e^{x\ -\ y}\ +\ 3\ x^2\ e^{-y}}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[\frac{dy}{dx}\ =\ e^{x\ -\ y}\ +\ 3\ x^2\ e^{-y}\]
\[\frac{dy}{dx}\ =\ e^x\ e^{-y}\ +\ 3\ x^2\ e^{-y}\]
\[\frac{dy}{dx}\ =\ e^{-y}(e^x\ +\ 3\ x^2)\]
\[\frac{dy}{e^{-y}}\ =\ (e^x\ +\ 3\ x^2)dx\]
\[e^{y}\ dy\ =\ (e^x\ +\ 3\ x^2)dx\]
\[Integrating\ on\ both\ sides\]
\[\int e^y\ dy\ =\ \int (e^x\ +\ 3\ x^2)\ dx\]
\[e^y\ =\ e^x\ +\ 3\ \frac{x^3}{3}\ +\ c\]
\[\boxed{e^y\ =\ e^x\ +\ x^3\ +\ c}\]

Example  7:

\[7.\ \color{red}{Solve:\ (1\ +\ e^x)\ sec^2\ y\ dy\ -\ e^x\ tan\ y\ dx\ =\ 0}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[(1\ +\ e^x)\ sec^2\ y\ dy\ -\ e^x\ tan\ y\ dx\ =\ 0\]
\[(1\ +\ e^x)\ sec^2\ y\ dy\ =\ \ e^x\ tan\ y\ dx\]
\[\frac{sec^2\ y}{tan\ y}\ dy\ =\ \frac{e^x}{1\ +\ e^x}\ dx\]
\[Integrating\ on\ both\ sides\]
\[\int \frac{sec^2\ y}{tan\ y}\ dy\ =\ \int \frac{e^x}{1\ +\ e^x}\ dx\]
\[put\ u\ =\ tan\ y\ \hspace{5cm}\ put\ z\ =\ 1\ +\ e^x\]
\[du\ =\ sec^2\ y\ dy\ \hspace{5cm}\ dz\ =\ e^x\ dx\]
\[\int \frac{du}{u}\ =\ \int \frac{dz}{z}\]
\[log\ u\ =\ log\ z\ +\ log\ c\]
\[log\ (tan\ y)\ =\ log\ (1\ +\ e^x)\ +\ log\ c\]
\[log\ (tan\ y)\ =\ log\ (1\ +\ e^x)\ c\]
\[tan\ y\ =\ (1\ +\ e^x)\ c\]