324 PRACTICAL MATHEMATICS 



dy o 



Example 2. Solve the equation ^ + y cos x = cos 3 x. 



IP das = I 



COS X 



= sin x 



sin x 



The integrating factor is e 



Hence ye sin * = f e sin * cos 3 x dx + c 



To find \e sin w cos 3 x dx, put z = sin x 

 Xhen ds = cos x dx, and cos 2 x = 1 



U 8in K cos 3 x dx = J e 2 (1 - z 2 ) ^2 



f ? dz - f 

 - {z 2 e* - 2 jze 2 dz} 



- 2(26* - je" dz) } 



and 8in K cos 3 



dz 

 e* 



= _ ^ (2 2 - 2z + 1) 

 = -e sinw (l - sin x} z 



Therefore ye sln * = c - e sin * (1 - sin *) 2 



and ^^-^-(i-sin^ 



dw . 



Example 3. Solve the equation x -^ + y = x 2 sm 

 Then + =a3sina? 



and 



= log e 03 



The integrating factor = e loge * 



Hence xy = U 2 sin x dx + c 



= a; 2 cos x + 2 la; cos x dx+c 



