Word Count : 410

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We are asked to solve the differential equation: $$ y'' + y = \sec^2 x $$ using the method of variation of parameters. --- Step 1: Solve the homogeneous equation The homogeneous equation is: $$ y_h'' + y_h = 0 $$ The characteristic equation is: $$ r^2 + 1 = 0 \implies r = \pm i $$ So the general solution of the homogeneous equation is: $$ y_h = C_1 \cos x + C_2 \sin x $$ --- Step 2: Variation of parameters The formula for variation of parameters for the nonhomogeneous equation $y'' + y = f(x)$ is: $$ y_p = __________ _______ __________ ________ ______ ____ ____ _______ ____.
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