Mumbai University > First Year Engineering > sem 2 > Applied Maths 2

Marks : 8

Year : 2015

Auxillary equation is $D^2 + 1 = 0\\ D=\pm i\\ Y_c=C_1\cos x+C_2\sin x\\ Let \space y_1=\cos x,y_2=\sin x$

$$\therefore W=\begin{vmatrix} y_1&y_2\\ y_1' & y_2'\\ \end{vmatrix} $$

$$ =\begin{vmatrix} \cos x&\sin x\\ -\sin x& \cos x\\ \end{vmatrix} $$

$=\cos^2x+\sin^2x\\ W=1\\ \therefore U=-\int \dfrac {y_2x}w\space dx\\ =-\int \dfrac {\sin x\sec x\tan x}1 dx\\ =-\int \tan^2 x dx\\ =-\int (\sec^2x-1) dx\\ =-\Big[\int \sec^2xdx-\int 1dx\Big]\\ U=-(\tan x-x)\\ =x-\tan x \\ And \space V=\int \dfrac {\cos x\sec x\tan x}1 dx\\ =\int \tan xdx\\ =\log|\sec x|\\ \therefore PI. , i.e. , y_p=uy_1+vy_2\\ y_p=(x-\tan)\cos x+\sin x\log\sec x\\ \therefore y=y_c+y_p\\ y=C_1\cos x+C_2\sin x+ x\cos x-\tan x\cos x+\sin x\log \sec x$