The integrating factor method, which was an effective method for solving first-order differential equations, is not a viable approach for solving second-order equstions. To see what happens, even for the simplest equation, consider the differential equation . Lagrange sought a function such that if one multiplied the left-hand side of bu , one would get where g(t)g(t) is to be determined. In this way, the given differential equation would be converted to , which could be integrated, giving the first-order equation which could be solved by first-order methods. (a) Differentate the right-hand side of and set the coefficients of y,y' and y'' equal to each other to find g(t). (b) Show that the integrating factor satisfies the second-order homogeneous equation called the adjoint equation of . In other words, althought it is possible to find an "integrating factor" for second-order differential equations, to find it one must solve a new second-order equation for the integrating factor μ, which might be every bit as hard as the original equation. (c) Show that the adjoint equation of the general second-order linear equation is the homogeneous equation .