Question
Using the method of variation of parameters, solve the equation
Answer :
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The given differential equation is [ \frac{d^2y}{dx^2} + a^2 y = \sec(ax) ] First, solve the complementary (homogeneous) equation: [ \frac{d^2y}{dx^2} + a^2 y = 0 ] The characteristic equation is [ r^2 + a^2 = 0 \implies r = \pm ai ] So the complementary solution is [ y_c = C_1 \cos(ax) + C_2 \sin(ax) ] Now, for variation of parameters, assume a particular solution of the form [ y_p = u_1(x)\cos(ax) + u_2(x)\sin(ax) ] where (u_1) and (u_2) satisfy [ u_1' \cos(ax) + u_2' \sin(ax)
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