The equation P(x)y + Qx)y + R(x)y = 0 %3D is said to be exact if...

   

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The equation P(x)y" + Qx)y + R(x)y = 0 %3D is said to be exact if it can be written in the form [P()yT+lfwy] = 0, where f(x) is to be determined in terms of P(x), Qx), and R(x). The latter equation can be integrated once immediately, resulting in a first order linear equation for y that can be solved as in Section 2.1. By equating the coefficients of the preceding equations and then eliminating f(x), show that a necessary condition for exactness is P"(x) - Qu) + R(x) = 0. It can be shown that this is also a sufficient condition. In each of Problems 42 through 45, use the result of Problem 41 to determine whether the given equation is exact. If it is, then solve the equation. 42. y" +xy +y =0 44. xy" - (cos x)y' + (sin x)y = 0, x> 0 43. y" +3xy +xy = 0 45. x*y" +xy - y = 0, x>0 The equation P(x)y" + Qx)y + R(x)y = 0 %3D is said to be exact if it can be written in the form [P()yT+lfwy] = 0, where f(x) is to be determined in terms of P(x), Qx), and R(x). The latter equation can be integrated once immediately, resulting in a first order linear equation for y that can be solved as in Section 2.1. By equating the coefficients of the preceding equations and then eliminating f(x), show that a necessary condition for exactness is P"(x) - Qu) + R(x) = 0. It can be shown that this is also a sufficient condition. In each of Problems 42 through 45, use the result of Problem 41 to determine whether the given equation is exact. If it is, then solve the equation. 42. y" +xy +y =0 44. xy" - (cos x)y' + (sin x)y = 0, x> 0 43. y" +3xy +xy = 0 45. x*y" +xy - y = 0, x>0

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