Homework SourseWhich of the following functions are solutions of the differ
Which of the following functions are solutions of the differential equation y^ll + y = sin x?
a.) y= sin x b.) y= cos x c.) y= 1/2x sin x d.) y= -1/2x cos x
a.) y= sin x b.) y= cos x c.) y= 1/2x sin x d.) y= -1/2x cos x
Solution
a) y = sinx; y\' = cosx; y\'\' = -sinx ........ y\'\' + y = -sinx + sinx = 0 ....... not solution b) y = cosx; y\' = -sinx; y\'\' = -cosx ...... y\'\' + y = -cosx + cosx = 0 ..... not solution c) y = 1/2 x sinx; y\' = 1/2 (x cosx + sinx); y\'\' = 1/2 (-x sinx + cosx + cosx) ... y\'\' + y = 1/2 (-x sinx + 2cosx) + 1/2 x sinx = cosx .................................... not solution d) y = -1/2 x cosx; y\' = -1/2(-x sinx + cosx); y\'\' = -1/2 (-x cosx - sinx - sinx) ... y\'\' + y = -1/2 (-x cosx - 2sinx) - 1/2 x cosx = sinx ...................................... OK Solution : d)
