(1 point) In general for a non-homogeneous problem y??+p(x)y?+q(x)y=f(x) assume that y1,y2 is a fundamental set...

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(1 point) In general for a non-homogeneous problemy??+p(x)y?+q(x)y=f(x) assume that y1,y2 is a fundamental set ofsolutions for the homogeneous problem y??+p(x)y?+q(x)y=0. Then theformula for the particular solution using the method of variationof parameters is yp=y1u1+y2u2 where u?1=?y2(x)f(x)W(x) andu?2=y1(x)f(x)W(x) where W(x) is the Wronskian given by thedeterminant W(x)=???y1(x)y?1(x)y2(x)y?2(x)??? So we haveu1=??y2(x)f(x)W(x)dx and u2=?y1(x)f(x)W(x)dx. NOTE When evaluatingthese indefinite integrals we take the arbitrary constant ofintegration to be zero. In other words we have the single integralformula yp(x)=y1(x)??y2(x)f(x)W(x)dx+y2(x)?y1(x)f(x)W(x)dx As aspecific example we consider the non-homogeneous problemy??+9y=sec2(3x) (1) The general solution of the homogeneous problem(called the complementary solution, yc=ay1+by2 ) is given in termsof a pair of linearly independent solutions, y1, y2. Here a and bare arbitrary constants. Find a fundamental set for y??+9y=0 andenter your results as a comma separated list ? BEWARE Notice thatthe above set does not require you to decide which function is tobe called y1 or y2 and normally the order you name them isirrelevant. But for the method of variation of parameters an ordermust be chosen and you need to stick to that order. In order tomore easily allow WeBWorK to grade your work I have selected aparticular order for y1 and y2. In order to ascertain the order youneed to use please enter a choice for y1= and if your answer ismarked as incorrect simply enter the other function from thecomplementary set. Once you get this box marked as correct then y2=. With this appropriate order we are now ready to apply the methodof variation of parameters. (2) For our particular problem we haveW(x)= u1=??y2(x)f(x)W(x)dx=? dx= u2=?y1(x)f(x)W(x)dx=? dx= Andcombining these results we arrive at yp= (3) Finally, using a and bfor the arbitrary constants in yc, the general solution can then bewritten as y=yc+yp=

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