\[Differential\ equations\ of\ the\ form\ \frac{dy}{dx}\ +\ P\ y\ =\ Q,\]\[Where\ P\ and\ Q\ are\ functions\ of\ x\ are\ called\ Linear\ Differential\ Equations\]
\[The\ solution\ of\ linear\ differential\ equation\ is\ given\ by\]
\[y\ (I.F)\ =\ \int Q\ (I.F)\ dx\ +\ c\]
\[Where\ I.F\ (Integrating\ Factor)\ =\ e^{\int P\ dx}\]
Example 1:
\[1.\ \color{red}{Find\ the\ integrating\ factor\ of\ \frac{dy}{dx}\ +\ \frac{y}{x}\ =\ x}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ \frac{1}{x}\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{\int \frac{1}{x}\ dx}\]
\[=\ e^{log\ x}\]
\[\boxed{I.F\ =\ x}\]
Example 2:
\[2.\ \color{red}{Find\ the\ integrating\ factor\ of\ \frac{dy}{dx}\ -\ \frac{3}{x}\ y\ =\ x^2}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ -\ \frac{3}{x}\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{-\ \int \frac{3}{x}\ dx}\]
\[=\ e^{-\ 3\int \frac{1}{x}\ dx}\]
\[=\ e^{-\ 3\ log\ x}\]
\[=\ e^{log\ x^{-3}}\]
\[=\ x^{-3}\]
\[\boxed{I.F\ =\ \frac{1}{x^3}}\]
Example 3:
\[3.\ \color{red}{Find\ the\ integrating\ factor\ of\ \frac{dy}{dx}\ +\ y\ cot\ x\ =\ x}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ cot\ x\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{ \int \frac{cos\ x}{sin\ x}\ dx}\]
\[put\ u\ =\ sin\ x\]
\[du\ =\ cos\ x\ dx\]
\[I.F\ =\ e^{ \int \frac{du}{u}}\]
\[=\ e^{log\ u}\]
\[=\ e^{log\ sin\ x}\]
\[=\ sin\ x\]
\[\boxed{I.F\ =\ sin\ x}\]
Example 4:
\[4.\ \color{red}{Solve:\ \frac{dy}{dx}\ +\ y\ tan\ x\ =\ 4x\ cos\ x}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ tan\ x\ \hspace{2cm}\ Q\ =\ 4x\ cos\ x\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{ \int \frac{sin\ x}{cos\ x}\ dx}\]
\[put\ u\ =\ cos\ x\]
\[du\ =\ -\ sin\ x\ dx\]
\[I.F\ =\ e^{ \int \frac{-du}{u}}\]
\[=\ e^{-\ log\ u}\]
\[=\ e^{log\ u^{-\ 1}}\]
\[=\ u^{-\ 1}\]
\[=\ \frac{1}{u}\]
\[\boxed{I.F\ =\ \frac{1}{cos\ x}}\]
\[The\ required\ solution\ is\]
\[y\ (I.F)\ =\ \int Q\ (I.F)\ dx\ +\ c\]
\[y\ (\frac{1}{cos\ x})\ =\ \int 4x\ cos\ x\ (\frac{1}{cos\ x})\ dx\ +\ c\]
\[y\ sec\ x\ =\ \int 4x\ \ dx\ +\ c\]
\[=\ 4\ \frac{x^2}{2}\ dx\ +\ c\]
\[\boxed{y\ sec\ x\ =\ 2\ x^2\ +\ c}\]
Example 5:
\[\color{red}{Solve:\ \frac{dy}{dx}\ +\ 2y\ tan\ x\ =\ e^{tan\ x}}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ 2\ tan\ x\ \hspace{2cm}\ Q\ =\ e^x\ cos\ x\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{ 2\int \frac{sin\ x}{cos\ x}\ dx}\]
\[put\ u\ =\ cos\ x\]
\[du\ =\ -\ sin\ x\ dx\]
\[I.F\ =\ e^{2 \int \frac{-du}{u}}\]
\[=\ e^{-\ 2\ log\ u}\]
\[=\ e^{log\ u^{-\ 2}}\]
\[=\ u^{-\ 2}\]
\[=\ \frac{1}{u^2}\]
\[ =\ \frac{1}{cos^2\ x}\]
\[\boxed{I.F\ =\ sec^2\ x}\]
\[The\ required\ solution\ is\]
\[y\ (I.F)\ =\ \int Q\ (I.F)\ dx\ +\ c\]
\[y\ sec^2\ x\ =\ \int e^{tan\ x}\ sec^2\ x\ dx\ +\ c\]
\[put\ u\ =\ tan\ x\]
\[du\ =\ sec^2\ x\ dx\]
\[y\ sec^2\ x\ =\ \int e^u\ \ du\ +\ c\]
\[=\ e^u\ +\ c\]
\[\boxed{y\ sec^2\ x\ =\ e^{tan\ x}\ +\ c}\]
Example 6:
\[\color{red}{Solve:\ \frac{dy}{dx}\ +\ y\ cot\ x\ =\ sin^3\ x}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ cot\ x\ \hspace{2cm}\ Q\ =\ sin^3\ x\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{ \int \frac{cos\ x}{sin\ x}\ dx}\]
\[put\ u\ =\ sin\ x\]
\[du\ =\ cos\ x\ dx\]
\[I.F\ =\ e^{ \int \frac{du}{u}}\]
\[=\ e^{log\ u}\]
\[=\ e^{log\ sin\ x}\]
\[=\ sin\ x\]
\[\boxed{I.F\ =\ sin\ x}\]
\[The\ required\ solution\ is\]
\[y\ (I.F)\ =\ \int Q\ (I.F)\ dx\ +\ c\]
\[y\ sin\ x\ =\ \int \ sin^3\ x\ sin\ x\ dx\ +\ c\]
\[=\ \int \ sin^4\ x\ dx\ +\ c\]
\[=\ \int \ (sin^2\ x)^2\ dx\ +\ c\]
\[=\ \int \ (\frac{1\ -\ cos\ 2x}{2})^2\ dx\ +\ c\]
\[=\ \frac{1}{4} [\int \ (1\ -\ cos\ 2x)^2\ dx]\ +\ c\]
\[=\ \frac{1}{4} [\int \ (1^2\ +\ (cos\ 2x)^2\ -\ 2(1)(cos\ 2x))\ dx]\ +\ c\]
\[=\ \frac{1}{4} [\int \ (1\ +\ cos^2\ 2x\ -\ 2\ cos\ 2x)\ dx]\ +\ c\]
\[W.K.T\ cos\ 2\ \theta\ =\ 2\ cos^2\ \theta\ -\ 1\]
\[cos\ 2\ \theta\ +\ 1\ =\ 2\ cos^2\ \theta\]
\[Replace\ \theta\ by\ 2x\]
\[cos\ 4\ x\ +\ 1\ =\ 2\ cos^2\ 2x\]
\[\frac{cos\ 4\ x\ +\ 1}{2}\ =\ cos^2\ 2x\]
\[y\ sin\ x\ =\ \frac{1}{4} [\int \ (1\ +\ \frac{cos\ 4\ x\ +\ 1}{2}\ -\ 2\ cos\ 2x)\ dx]\ +\ c\]
\[=\ \frac{1}{4} [\int \ 1\ dx\ +\ \frac{1}{2} \int cos\ 4\ x\ +\ \frac{1}{2} \int \ 1\ dx\ -\ 2\ \int cos\ 2x\ dx]\ +\ c\]
\[=\ \frac{1}{4} [\int \ 1\ dx\ +\ \frac{1}{2} \int cos\ 4\ x\ +\ \frac{1}{2} \int \ 1\ dx\ -\ 2\ \int cos\ 2x\ dx]\ +\ c\]
\[=\ \frac{1}{4} [x\ +\ \frac{1}{2}\ \frac{sin\ 4\ x}{4}\ +\ \frac{1}{2}\ x\ -\ 2\ \frac{sin\ 2\ x}{2}]\ +\ c\]
\[=\ \frac{1}{4} [\frac{3}{2}\ x\ +\ \frac{sin\ 4\ x}{8}\ -\ sin\ 2\ x]\ +\ c\]
\[\boxed{y\ sin\ x\ =\ \frac{3}{8}\ x\ +\ \frac{sin\ 4\ x}{32}\ -\ \frac{sin\ 2\ x}{4}\ +\ c}\]
Example 7:
\[\color{red}{Solve:\ \frac{dy}{dx}\ -\ \frac{2x}{1\ +\ x^2}\ y\ =\ (1\ +\ x^2)}\ \hspace{15cm}\]
\[\color {blue}{Soln:}\ \hspace{20cm}\]
\[Compare\ with\ \frac{dy}{dx}\ +\ P\ y\ =\ Q\]
\[Here\ P\ =\ -\ \frac{2x}{1\ +\ x^2}\ \hspace{2cm}\ Q\ =\ 1\ +\ x^2\]
\[I.F\ =\ e^{\int P\ dx}\]
\[=\ e^{ -\ \int \frac{2\ x}{1\ +\ x^2}\ dx}\]
\[put\ u\ =\ 1\ +\ x^2\]
\[du\ =\ 2\ x\ dx\]
\[I.F\ =\ e^{ \int \frac{-du}{u}}\]
\[=\ e^{-\ log\ u}\]
\[=\ e^{log\ u^{-\ 1}}\]
\[=\ u^{-\ 1}\]
\[=\ \frac{1}{u}\]
\[\boxed{I.F\ =\ \frac{1}{1\ +\ x^2}}\]
\[The\ required\ solution\ is\]
\[y\ (I.F)\ =\ \int Q\ (I.F)\ dx\ +\ c\]
\[y\ (\frac{1}{1\ +\ x^2})\ =\ \int (1\ +\ x^2)\ (\frac{1}{1\ +\ x^2})\ dx\ +\ c\]
\[\frac{y}{1\ +\ x^2}\ =\ \int 1\ dx\ +\ c\]
\[\boxed{\frac{y}{1\ +\ x^2}\ =\ x\ +\ c}\]
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