Word Count : 1055

---

For the first equation: [ \frac{d^2y}{dx^2} - y = \frac{2}{1 + e^x} ] Step 1: Solve the complementary (homogeneous) equation [ \frac{d^2y}{dx^2} - y = 0 ] The characteristic equation is: [ r^2 - 1 = 0 \implies r = \pm 1 ] So the complementary solution is: [ y_c = C_1 e^x + C_2 e^{-x} ] Step 2: Use variation of parameters for the particular solution Let (y_p = u_1 e^x + u_2 e^{-x}), where [ u_1' e^x + u_2' e^{-x} = 0 \quad \text{and} \quad u_1' e^x - u_2' e^{-x} = \frac{2}{1 + e^x} ] From the first equation: [ u_1' e^x + u_2' e^{-x} = 0 \implies u_2' = -u_1' e^{2x} ] Substitute into the second equation: [ u_1' e^x - (-u_1' e^{2x}) e^{-x} = u_1' e^x + u_1' e^x = 2 u_1' e^x = \frac{2}{1 + e^x} \implies u_1' = \frac{1}{e^x (1 + e^x)} = \frac{1}{e^x + e^{2x}} = \frac{1}{e^x(1 + e^x)} ] [ u_1' = \frac{1}{e^x(1 + e^x)} = \frac{1}{e^x (1 + e^x)} ] [ u_1 = \int \frac{dx}{e^x (1 + e^x)} = \int \frac{e^{-x} dx}{1 + e^x} ] Let (t = e^x \implies dt = e^x dx \implies dx = \frac{dt}{t}) [ u_1 = \int \frac{e^{-x}}{1 + e^x} dx = \int \frac{1/t}{1 + t} \cdot __________ _________ _____ __________ ______ _____ __________.
____ _____ ______ ________ _______ __________ ____ _________ ______ __________.
______ __________ _____ ___ _____ ______.
______ _____ ____ _________ _________ _______ __________ ___ ______ ______.
_____ ____ _______ ____ ____ ___.
____ _________ __________ _________ _________ ______ ________ _____ ______ ______ ______.
___ _______ _____ ______ _________ _______ ______ ___ _________ _____ ___ ___.
________ ___ _______ _________ ___ ______ ____ _________ ______ ___ ____.
________ ___ ___ ________ __________ _________ _______ __________ __________ _________.
_____ ______ ________ _______ _______ __________ _______ ________ __________ ____ _________.
___ ____ ___ ____ ___ _____ ______ ____ _______ ________ ________.
__________ _____ _____ __________ _______ __________ _______.
_______ _____ _______ _______ ________.
_____ __________ ____ ____ __________ _________ ______ _____ __________ ___ ___.
_________ ______ ______ ____ _______ ___ ___ _________ ___ _____.
_______ _____ _________ __________ ______ ______ _________ _________ _______ ______.
__________ ________ _________ _____ ______ _________ __________ _________ _________ ________ _______.
_______ _______ ___ __________ ________ _______.
_________ ____ ______ _________ ______ ____ ________ ____ _______ __________.
____ __________ __________ _________ _____ _____ ___ ______ _______ _______.
_________ _______ ______ _________ ________ ________ ______ ___.
__________ __________ ________ ____ ______ _________ ____ _____ __________ _______ ________.
____ ___ _________ ___ ____ __________ ________ __________ _________ _____ _______ ___.
________ ___ ________ _____ ____ ___ ________ ________ _____.
________ ____ ___ ___ ______ _____ ___ ___ ____ _____ _____ ________.
___ ____ _______ _______ ________ ______ _______ ______.
____ ___ _________ ________ ________ ________.
_____ _________ _____ ________ _________ _____.
__________ ____ _______ _______ _________ __________ ________ _______.
_________ _______ ________ ____ __________.
____ _________ ________ ______ __________ __________ ___ __________.
_________ ___ ____ ____ _______.
________ _________ _______ ____ _____ ________ ____ ________ _________ _________.
__________ __________ _________ __________ __________ _______ ____.
______ ______ _____ _________ ______.
_________ _________ ______ __________ _______ _________ ___ __________.
_________ ________ _______ _________ ________ _______ _____ __________ __________.
________ _______ ____ ____ ___.
_________ _____ _______ ________ ______ _______ ______ ____ ___ ___.
________ ________ _______ ____ _____ ____ _______ _________.
_____ __________ ______ __________ ___.
_________ _____ _________ ___ ______ _____ ___ ___ __________ ___ ______ _______.
___ _________ _______ ______ ___ ___ ________.
______ ________ ________ ______ _____ _________ ____ ___ __________ _________ _______ ________.
________ ________ ________ ___ __________ _______.
_________ ________ _________ _________ _____ _____ ____ ___ ________ _____.
_______ ____ ____ ______ ______ ____ ___ _________.
_______ _______ _________ ___ _______ ____ ____ __________.
__________ _____ __________ ______ __________ __________ ____ __________ ________ ___ ____ _________.
___ ________ _________ ____ _________ _____.
______ _______ __________ _______ ________ __________ ________ _________.
________ __________ ____ _______ _________ ___ ____ __________ _____.
________ ______ ______ ____ __________ ____ ________ ________.
_______ __________ ___ _________ ________ ______ ________ ___ _________ ___.
_____ ______ _________ ________ _________ ______ ______ ________ _______ __________ ________ _____.
___ ________ ________ _____ ____ _________ ________.
__________ ____ ________ ________ ________ ________.
___ ___ ___ __________ _________ ___.
______ __________ ______ ____ _______ ____ ____ __________.
____ ______ _______ ______ _______ ______ ____ _____ ___ ____.
_________ _______ _____ __________ ________ ___ _________.
_______ ________ ___ _________ _______ __________.
_______ _______ ___ __________ _______ _________ _________ __________ ____ __________ ___.
____ _____ _____ ____ _____.
_____ __________ ______ __________ __________ _______.
_____ __________ ______ ________ _____ __________ ______ ____ _______ _____ _________ __________.
_____ ___ ________ ____ _______ _________ _____.
_________ _______ _____ ________ __________.
____ _______ __________ _____ ______ ____ _______ ____ _____ _____ ________ _____.
_________ _______ ________ ________ _____ _____ __________.
_________ ________ ____ ____ _____ _________.
________ ____ __________ ______ ____ ______ _____.
___ __________ _________ _____ ______ _________.
_________ ______ __________ ______ ____ ________ _________.
____ _______ ________ __________ ____ _______ _____ ______ __________.
_____ ___ _________ _____ ________ ____ ____ __________ ____ ___.
______ _______ _________ _________ _____ _______ ________ ________ ___.
_______ __________ _____ ___ _________ _______ __________.
_____ ______ ___ ____ _____.
_______ ____ ____ ______ ____.
__________ _______ ________ ______ ___ ____ ________ ____ _____.
____ _____ ____ _______ _____ _________ ______ ________ ________.
______ _____ ________ ___ _____ ______ ____ _________ ___ ___ _____ __________.
_________ __________ _______ _________ ____ ___ _______ ______.
___ _____ ___ ____ ____ ___ ___ ________ ___ _____ _________.
______ _________ ________ ________ ______ _________ ________ _________ _________.
_________ ______ _________ ________ ____ ______ __________ ________ _________ _________ __________ ___.
________ ____ ________ _____ ________ _________ ________ __________ ____.
__________ ______ __________ ________ ________ _____ ______ ________ ________ ______.
___ ____ ______ _______ _________ ____ ___ _______ _____ __________.
___ __________ ______ ___ _________ ________ ___ ____ __________ _______ ______ _____.
_____ _____ ___ ________ _______ __________ ______ __________ ____.
______ __________ ____ ________ _____ ______ _________.
________ _________ _________ _______ _________ _________ _______.
___ _________ _______ ______ _____.
_________ ______ _________ ______ ___ _________ ________.
_______ _____ _________ _________ _____ ______ __________.
____ _______ ___ _______ ________ ___ ______ _______ _______ __________ _____.
________ __________ ___ ______ _______ _______ ____ ______ ________ __________ __________.
________ ____ ____ _____ __________.
Get Full Answer on WhatsApp