Word Count : 620

---

We are tasked with finding the general solution to the differential equation: \[ y^{(iv)} - 2y'' + 2y'' = 3e^{-x} + 2xe^{-x} + e^{-x} \sin x. \] ### Step 1: Simplify the differential equation Notice that the term \( -2y'' + 2y'' \) cancels out, leaving us with the following equation: \[ y^{(iv)} = 3e^{-x} + 2xe^{-x} + e^{-x} \sin x. \] ### Step 2: Solve the homogeneous equation The homogeneous equation associated with this is: \[ y^{(iv)} = 0. \] The characteristic equation for the homogeneous part is: \[ r^4 = 0. \] Thus, \( r = 0 \) is a repeated root of multiplicity 4. The general solution to the homogeneous equation is: \[ y_h(x) = C_1 + C_2 x + C_3 ________ _______ __________ _____ _____ ___.
_________ _______ _______ _____ _______ ____ ________ _______ ________ ____ __________ ___.
__________ __________ _______ ______ _____ __________ _______ ______ _______ ________.
_____ _______ _________ ___ ________ ________.
_____ _______ ______ _____ __________ ____ ___ _______ _______ __________ ______.
_______ _________ _________ _________ __________ ____ _____ __________ ______.
_________ ______ ____ __________ ___ ________ ___ ___ _______ ________ __________.
_____ _____ _______ ____ ______ ___ ______ ___.
__________ ____ _________ _______ _______ ___ ________ ____ _________ ___.
____ _____ ___ __________ _________.
_____ _________ ____ _______ __________ _______.
__________ ________ ________ ______ __________ _______ ______ ____ ___ __________ _________ _____.
_____ _______ ___ _______ ___ __________.
__________ __________ _______ ________ _____ __________ _________ __________ __________ ________ ___ _______.
___ ______ __________ _________ _______ __________ _____ ______ ________.
___ ______ _____ ______ ______ ________ ______ __________ _____ __________ __________ _________.
____ ________ ______ ______ _______ ______ _____ _____ _______ ______ ______ __________.
________ ___ ____ _____ ________ __________ ______ __________ ____ ______.
_____ ___ ___ _________ ____ ___ _____ _______ ______ _________ _________.
_______ _______ __________ _______ ____ ____ _________ _____.
____ ________ ______ _______ __________ _____ _________.
______ _________ _______ ___ _________ ____ ________ ____ ________ ___ _______.
____ _________ _________ _________ ______ __________ _____.
_____ ____ _______ ___ ___ __________ ____ _______.
____ _____ ________ ___ ___ _____ ______ _____ _____.
_____ ________ ___ ________ ______ ____ ________ _________ ___ ______.
__________ ___ ____ _______ _____ ________.
____ ______ ________ _______ ____ ________ ______ _____ __________ __________.
_______ _______ ____ _______ _____ ______ ________ ______ ______ _________ _____ _____.
___ ___ __________ __________ __________ __________.
_____ __________ ______ ___ ______ ___ ____ _____ _________ ____ ____.
_____ ________ __________ _________ ___ _______ _______ ____ _____ ______ ___ _____.
_______ ___ ________ _________ _________ __________ _________ __________ ______ _______ _______ __________.
___ _____ ________ ___ _________ _______ ____ ___ _____ ______.
___ _____ __________ _____ ____ _________.
________ _____ __________ ______ _________ __________ ____.
_______ __________ _______ __________ _____ _______ _______ ___ ___.
______ ____ ____ _________ _____ _______.
_____ ______ _____ _______ ____ _______ ______ ____ _____ _______.
__________ ________ __________ ___ ________ ___ ___ __________ ______ ___ _______ ____.
__________ __________ ______ ________ ________ ____ ____.
______ _______ ____ ____ _________ ________ ________ ___ _________ ___.
____ ___ _________ ___ ____ _______ _____ _____ __________ ___.
____ __________ _________ _______ ______ _________ ________ _________ ___.
______ ____ _____ ____ ______ ____ ____ __________ ______ ___ _____ _________.
___ _________ _____ ______ ________ _______ ____ ______ _____ _____.
_________ ______ _____ _________ ________ _____ _________ _____ __________.
_______ _________ _______ ________ __________ ______ ______ _______ ______ ___.
_____ _______ ___ __________ ________ _____ __________ ___.
___ _______ ________ __________ __________ ____ _________ __________ __________ ______.
_______ ________ ________ ______ _____ ____ _________ __________ ________ _______ ________.
___ ________ ___ __________ __________ ______ _________ __________.
___ _________ ____ __________ _________ __________ _______.
____ _________ ________ ______ ____ ___ ______ _______.
Get Full Answer on WhatsApp