SUCCESSIVE DIFFERENTIATION (Text)

\[\color {royalblue} {Notation\ of\ successive\ derivatives}:\ \hspace{20cm}\]
\[1.\ \frac{dy}{dx}\ =\ y_1\ =\ f^\prime(x)\ =\ D(y)\]
\[2.\ \frac{d^2y}{dx^2}\ =\ y_2\ =\ f^{\prime \prime}(x)\ =\ D^2(y)\]
\[\color {royalblue} {Differential Equation}:\ \hspace{20cm}\]
\[\color {royalblue} {Definition}:\ \hspace{20cm}\]
\[An\ equation\ containing\ differential\ coefficients\ is\ called\ differential\ equation.\]
\[\color {royalblue} {Examples}:\ \hspace{20cm}\]
\[1.\ \frac{dy}{dx}\ +\ y\ =\ x\]
\[2.\ x\frac{d^2y}{dx^2}\ +\ 2\ \frac{dy}{dx}\ =\ xy\]
\[\color {royalblue} {Order\ of\ the\ differential\ equation}:\ \hspace{20cm}\]
\[The\ order\ of\ the\ highest\ derivative\ in\ the\ differential\ equation\]\[is\ called\ order\ of\ the\ differential\ equation.\]
\[\color {royalblue} {Degree\ of\ the\ differential\ equation}:\ \hspace{20cm}\]
\[The\ power\ of\ the\ highest\ derivative\ in\ the\ differential\ equation\]\[is\ called\ the\ degree\ of\ the\ differential\ equation.\]
\[\color {royalblue} {Example}:\ \hspace{20cm}\]
\[Find\ the\ order\ and\ degree\ of\ 7(\frac{d^2y}{dx^2})^2\ +\ 5\ (\frac{dy}{dx})^5\ +\ 7\ y\ =\ Sin\ x\ \hspace{15cm}\]
\[\color {black}{Solution:}\ order\ =\ 2,\ degree\ =\ 2\ \hspace{15cm}\]
\[\color {royalblue} {Problems}:\ \hspace{20cm}\]
\[Ex\ 1:\ Form\ the\ differential\ equation\ by\ eliminating\ the\ constant\ ‘a’\ in\ x^2\ +\ y^2\ =\ a^2\ \hspace{15cm}\]
\[\color {black}{Solution:}\ x^2\ +\ y^2\ =\ a^2\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}( x^2)\ +\ \frac{d}{dx}( y^2)\ =\ \frac{d}{dx}( a^2)\ \hspace{10cm}\]
\[2\ x\ +\ 2\ y\ \frac{dy}{dx}\ =\ 0\ \hspace{10cm}\]
\[2\ y\ \frac{dy}{dx}\ =\ -\ 2\ x\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ -\ \frac{x}{y}\ \hspace{10cm}\]
\[Ex\ 2:\ Find\ \frac{d^2y}{dx^2}\ if\ y\ =\ e^{2x}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ y\ =\ e^{2x}\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( e^{2x})\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ 2\ e^{2x}\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(\frac{dy}{dx})\ =\ \frac{d}{dx}( 2\ e^{2x})\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 4\ e^{2x}\ \hspace{10cm}\]
\[Ex\ 3:\ Find\ \frac{d^2y}{dx^2}\ if\ y\ =\ Sec\ x\ \hspace{15cm}\]
\[\color {black}{Solution:}\ y\ =\ Sec\ x\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( Sec\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ Sec\ x\ Tan\ x\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(\frac{dy}{dx})\ =\ \frac{d}{dx}( Sec\ x\ Tan\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ Sec\ x\ \frac{d}{dx}(Tan\ x)\ +\ Tan\ x\ \frac{d}{dx}(Sec\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ Sec\ x\ Sec^2\ x\ +\ Tan\ x\ Sec\ x\ Tan\ x\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ Sec^3\ x\ +\ Tan^2\ x\ Sec\ x\ \hspace{10cm}\]
\[Ex\ 4:\ Find\ \frac{d^2y}{dx^2}\ if\ y\ =\ Sin\ 3\ x\ \hspace{15cm}\]
\[\color {black}{Solution:}\ y\ =\ Sin\ 3\ x\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( Sin\ 3\ x)\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ Cos\ 3\ x\ \frac{d}{dx}(3\ x)\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ Cos\ 3\ x\ 3(1)\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ 3\ Cos\ 3\ x\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(\frac{dy}{dx})\ =\ \frac{d}{dx}( 3\ Cos\ 3\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 3\ \frac{d}{dx}(Cos\ 3\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ 3\ (-\ Sin\ 3\ x\ 3(1))\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ -\ 9\ Sin\ 3\ x\ \hspace{10cm}\]
\[Ex\ 5:\ If\ y\ =\ x^2\ Sin\ x,\ find\ \frac{d^2y}{dx^2}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ y\ =\ x^2\ Sin\ x\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( x^2\ Sin\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^2\ \frac{d}{dx}(Sin\ x)\ +\ Sin\ x\ \frac{d}{dx}(x^2)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^2\ Cos\ x\ +\ Sin\ x\ 2\ x\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^2\ Cos\ x\ +\ 2\ x\ Sin\ x\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(\frac{dy}{dx})\ =\ \frac{d}{dx}(x^2\ Cos\ x\ +\ 2\ x\ Sin\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ \frac{d}{dx}(x^2\ Cos\ x)\ +\ 2\ \frac{d}{dx}(x\ Sin\ x)\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ x^2\ \frac{d}{dx}(Cos\ x)\ +\ Cos\ x\ \frac{d}{dx}(x^2)\ +\ 2[x\ \frac{d}{dx}(Sin\ x)\ +\ Sin\ x\ \frac{d}{dx}(x)]\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ x^2\ (-\ Sin\ x)\ +\ Cos\ x\ (2\ x)\ +\ 2[x\ Cos\ x\ +\ Sin\ x\ (1)]\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ -\ x^2\ Sin\ x\ +\ 2\ x\ Cos\ x\ +\ 2\ x\ Cos\ x\ +\ 2\ Sin\ x\ \hspace{10cm}\]
\[\frac{d^2y}{dx^2}\ =\ -\ x^2\ Sin\ x\ +\ 4\ x\ Cos\ x\ +\ 2\ Sin\ x\ \hspace{10cm}\]
\[Ex\ 6:\ If\ y\ =\ x^2\ Sin\ x,\ prove\ that\ x^2\ y_2\ -\ 4\ x\ y_1\ +\ (x^2\ +\ 6)y\ =\ 0\ \hspace{15cm}\]
\[(OR)\]
\[If\ y\ =\ x^2\ Sin\ x,\ prove\ that\ x^2\ \frac{d^2y}{dx^2}\ -\ 4\ x\ \frac{dy}{dx}\ +\ (x^2\ +\ 6)y\ =\ 0\ \hspace{15cm}\]
\[\color {black}{Solution:}\ y\ =\ x^2\ Sin\ x\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( x^2\ Sin\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ x^2\ \frac{d}{dx}(Sin\ x)\ +\ Sin\ x\ \frac{d}{dx}(x^2)\ \hspace{10cm}\]
\[y_1\ =\ x^2\ Cos\ x\ +\ Sin\ x\ 2\ x\ \hspace{10cm}\]
\[y_1\ =\ x^2\ Cos\ x\ +\ 2\ x\ Sin\ x\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y_1)\ =\ \frac{d}{dx}(x^2\ Cos\ x\ +\ 2\ x\ Sin\ x)\ \hspace{10cm}\]
\[y_2\ =\ \frac{d}{dx}(x^2\ Cos\ x)\ +\ 2\ \frac{d}{dx}(x\ Sin\ x)\ \hspace{10cm}\]
\[y_2\ =\ x^2\ \frac{d}{dx}(Cos\ x)\ +\ Cos\ x\ \frac{d}{dx}(x^2)\ +\ 2[x\ \frac{d}{dx}(Sin\ x)\ +\ Sin\ x\ \frac{d}{dx}(x)]\ \hspace{10cm}\]
\[y_2\ =\ x^2\ (-\ Sin\ x)\ +\ Cos\ x\ (2\ x)\ +\ 2[x\ Cos\ x\ +\ Sin\ x\ (1)]\ \hspace{10cm}\]
\[y_2\ =\ -\ x^2\ Sin\ x\ +\ 2\ x\ Cos\ x\ +\ 2 x\ Cos\ x\ +\ Sin\ x\ \hspace{10cm}\]
\[y_2\ =\ -\ x^2\ Sin\ x\ +\ 4\ x\ Cos\ x\ +\ Sin\ x\ \hspace{10cm}\]
\[L.\ H.\ S\ =\ -\ x^2\ y_2\ -\ 4\ x\ y_1\ +\ (x^2\ +\ 6)y\ \hspace{10cm}\]
\[=\ -\ x^2\ (-\ x^2\ Sin\ x\ +\ 4\ x\ Cos\ x\ +\ Sin\ x)\ -\ 4\ x\ (x^2\ Cos\ x\ +\ 2\ x\ Sin\ x)\ +\ (x^2\ +\ 6)(x^2\ Sin\ x)\ \hspace{10cm}\]
\[=\ x^4\ Sin\ x\ -\ 4\ x^3\ Cos\ x\ -\ x^2\ Sin\ x)\ -\ 4\ x^3\ Cos\ x\ -\ 8\ x^2\ Sin\ x)\ +\ x^4\ Sin\ x+\ 6\ x^2\ Sin\ x\ \hspace{10cm}\]
\[=\ 0\ =\ R.\ H.\ S\ \hspace{10cm}\]
\[Ex\ 6:\ If\ y\ =\ e^x\ Sin\ x,\ prove\ that\ y_2\ -\ 2\ y_1\ +\ 2\ y\ =\ 0\ \hspace{15cm}\]
\[\color {black}{Solution:}\ y\ =\ e^x\ Sin\ x\ —–\ (1)\ \hspace{15cm}\]
\[Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( e^x\ Sin\ x)\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ e^x\ \frac{d}{dx}(Sin\ x)\ +\ Sin\ x\ \frac{d}{dx}(e^x)\ \hspace{10cm}\]
\[y_1\ =\ e^x\ Cos\ x\ +\ y\ using(1)\ —-\ (2)\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(y_1)\ =\ \frac{d}{dx}(e^x\ Cos\ x\ +\ y)\ \hspace{10cm}\]
\[y_2\ =\ \frac{d}{dx}(e^x\ Cos\ x)\ +\ \frac{d}{dx}(y)\ \hspace{10cm}\]
\[y_2\ =\ e^x\ \frac{d}{dx}(Cos\ x)\ +\ Cos\ x\ \frac{d}{dx}(e^x)\ +\ y_1\ \hspace{10cm}\]
\[y_2\ =\ e^x\ (-\ Sin\ x)\ +\ Cos\ x\ (e^x)\ +\ y_1\ \hspace{10cm}\]
\[y_2\ -\ y_1\ =\ -\ y\ +\ (y_1\ -\ y)\ using(2)\ \hspace{10cm}\]
\[y_2\ -\ y_1\ =\ -\ y\ +\ y_1\ -\ y\ \hspace{10cm}\]
\[y_2\ -\ y_1\ =\ -\ 2\ y\ +\ y_1\ \hspace{10cm}\]
\[y_2\ -\ y_1\ +\ 2\ y\ -\ y_1\ =\ 0\ \hspace{10cm}\]
\[y_2\ -\ 2\ y_1\ +\ 2\ y\ =\ 0\ \hspace{10cm}\]
\[Ex\ 7:\ If\ y\ =\ a\ Cos(log\ x)\ +\ b\ Sin\ (log\ x),\ prove\ that\ x^2\ y_2\ +\ x\ y_1\ +\ y\ =\ 0\ \hspace{10cm}\]
\[\color {black}{Solution:}\ y\ =\ a\ Cos(log\ x)\ +\ b\ Sin\ (log\ x)\ —–\ (1)\ \hspace{15cm}\]
\[Differentiate\ both\ sides\ w.\ r.\ t.\ x\ \hspace{10cm}\]
\[\frac{d}{dx}(y)\ =\ \frac{d}{dx}( a\ Cos(log\ x)\ +\ b\ Sin\ (log\ x))\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ a\ \frac{d}{dx}(Cos(log\ x))\ +\ b\ \frac{d}{dx}(Sin\ (log\ x))\ \hspace{10cm}\]
\[\frac{dy}{dx}\ =\ a\ (-\ Sin(log\ x))\ \frac{d}{dx}(log\ x)\ +\ b\ Cos(log\ x)\ \frac{d}{dx}(log\ x)\ \hspace{10cm}\]
\[y_1\ =\ -\ a\ Sin(log\ x)\ \frac{1}{x}\ +\ b\ Cos(log\ x)\ \frac{1}{x}\ \hspace{10cm}\]
\[y_1\ =\ \frac{1}{x}[-\ a\ Sin(log\ x)\ +\ b\ Cos(log\ x)]\ \hspace{10cm}\]
\[x\ y_1\ =\ -\ a\ Sin(log\ x)\ +\ b\ Cos(log\ x)\ \hspace{10cm}\]
\[Again\ Differentiate\ w.\ r.\ t.\ x\ on\ both\ sides\ \hspace{10cm}\]
\[\frac{d}{dx}(x\ y_1)\ =\ \frac{d}{dx}(-\ a\ Sin(log\ x)\ +\ b\ Cos(log\ x))\ \hspace{10cm}\]
\[x\ \frac{d}{dx}(y_1)\ +\ y_1\ \frac{d}{dx}(x)\ =\ -\ a\ \frac{d}{dx}(Sin(log\ x)\ +\ b\ \frac{d}{dx}(Cos(log\ x)\ \hspace{10cm}\]
\[x\ y_2\ +\ y_1\ (1)\ =\ -\ a\ (Cos(log\ x))\ \frac{d}{dx}(log\ x)\ +\ b\ (-\ Sin(log\ x)\ \frac{d}{dx}(log\ x)\ \hspace{10cm}\]
\[x\ y_2\ +\ y_1\ =\ -\ a\ Sin(log\ x)\ \frac{1}{x}\ -\ b\ Sin(log\ x)\ \frac{1}{x}\ \hspace{10cm}\]
\[x\ y_2\ +\ y_1\ =\ \frac{-\ 1}{x}[a\ Sin(log\ x)\ +\ b\ Cos(log\ x)]\ \hspace{10cm}\]
\[x\ y_2\ +\ y_1\ =\ \frac{-\ 1}{x}\ [y]\ ——-\ Using\ (1)\ \hspace{10cm}\]
\[x(x\ y_2\ +\ y_1)\ =\ y\ \hspace{10cm}\]
\[x^2\ y_2\ +\ x\ y_1\ =\ y\ \hspace{10cm}\]
\[x^2\ y_2\ +\ x\ y_1\ -\ y\ =\ 0\ \hspace{10cm}\]