Differential Equation: Prove identities.
1. Given xy  a sin 2x , prove that x
d2y
dy
 2  4 xy  0 .
2
dx
dx
2
d2y
 dy 
2. Given y  1  cos x , prove that 2 y 2  2    y 2  1 .
dx
 dx 
2
d2y
dy
 4 x  (2  x 2 ) y  0 .
2
dx
dx
2
d y
dy
4. Given y  e x sin x , show that
 2  2y  0.
2
dx
dx
2
d y
dy
5. Given y  e3 x sin 4 x , show that
 6  25 y  0 .
2
dx
dx
2
d y
dy
6. Given y  ae  mx cos px , show that
 2m  (m2  p 2 ) y  0 .
2
dx
dx
3. Given x 2 y  4cos x , prove that x 2