Let y cos ( kt ) ( a ) Find the values of k such that y satisfies the differential equation 1 6 y 8 1 y ( Enter your answers as a comma separated list ) k ( b ) For the values of k found in part ( a ) , show that every member of the family of functions y A sin ( kt ) B cos ( kt ) is also a solution of the differential equation We begin by calculating the following y A sin ( kt ) B cos ( kt ) y Ak cos ( kt ) Bk sin ( kt ) y Note that the given differential equation 1 6 y 8 1 y is equivalent to 1 6 y 8 1 y Now, substituting the expressions for y and y above and simplifying, we have LHS 1 6 y 8 1 y 1 6 8 1 ( A sin ( kt ) B cos ( kt ) ) 1 6 1 6 Bk 2 cos ( kt ) 8 1 A sin ( kt ) 8 1 B cos ( kt ) ( 8 1 1 6 k 2 ) ( 8 1 1 6 k 2 ) B cos ( kt ) 0 since for all value of k found above, k 2

Question

Let y   cos ( kt )   ( a ) Find the values of k such that y satisfies the differential equation 1 6 y   8 1 y   ( Enter your answers as a comma   separated list  ) k   ( b ) For the values of k found in part ( a ) , show that every member of the family of functions y   A sin ( kt )   B cos ( kt ) is also a solution of the differential equation  We begin by calculating the following  y   A sin ( kt )   B cos ( kt ) y   Ak cos ( kt ) Bk sin ( kt ) y   Note that the given differential equation 1 6 y   8 1 y is equivalent to 1 6 y   8 1 y     Now, substituting the expressions for y and y above and simplifying, we have LHS   1 6 y   8 1 y   1 6   8 1 ( A sin ( kt )   B cos ( kt ) )   1 6 1 6 Bk 2 cos ( kt )   8 1 A sin ( kt )   8 1 B cos ( kt )   ( 8 1 1 6 k 2 )   ( 8 1 1 6 k 2 ) B cos ( kt )   0 since for all value of k found above, k 2

HelloExperts · Accepted Answer

The Answer is in the image, click to view...

Let y = cos ( kt ) . ( a ) Find the values of k such that y satisfies the differential equation 1 6

Step by Step Solution

1 Understanding Responsive Design Principles

Related Calculus Of Variations Questions