Solve using variation of parameters

Question

Use variation of parameters to solve the general solution of the following:

(D² + 1)y= sec x cscx

Answer 1

The complementary solution is y_c= c₁cosx +c₂sinx.

Let the particular solution be of the form

y_p = u₁ y₁+ u₂ y₂ where y₁= cosx and y₂= sinx

Then solving the system

u₁' y₁ + u₂' y₂ = 0

u₁' y₁'+ u₂' y₂' ₌ secx cscx

we get u₂' =− ( cscx) ⇒ u₂= - ln |cscx +cotx |

and u₁' = −secx ⇒ u₁ = −ln| secx + tanx |

Then the final solution is y = y_c +y_p = c₁cosx +c₂sinx − cosx ln| secx + tanx | − sinx ln |cscx +cotx |