Word Count : 523

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For the differential equation $y'' + y = \sec^2 x$, we first solve the homogeneous equation: Homogeneous equation: $$ y_h'' + y_h = 0 $$ Characteristic equation: $$ r^2 + 1 = 0 \implies r = \pm i $$ So the complementary solution is: $$ y_c = C_1 \cos x + C_2 \sin x $$ Now, for the particular solution $y_p$ using variation of parameters: $$ y_p = u_1(x)\cos x + u_2(x)\sin x $$ Where $$ u_1' = -\frac{y_2 f(x)}{W}, \quad u_2' = \frac{y_1 f(x)}{W} $$ Here, ___ _____ ____ ______ ________ ___ _______ _________ ________ _______ ________ ___.
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