Lagrange's Adjoint Equation: The integrating factor method, which was an effective method for solving first order differentia1 equations, is not a viable approach for solving second-order equations. To see what happens even for the simplest equation, consider the differential equation
y" + 3y' + 2y = f(t)
Lagrange sought a function μ(t) such that if one multiplied the left-hand side of (18) by μ(t). one would get
μ(t)[y" + y' + y] = d/dt [μ(t)y + g(t)y]
Where g(t) is to be determined. In this way, the given differential equation would be converted to
d/dt [μ (t)y' + g(t)y] = μ(1) f (01).
Which could be integrated, giving the first order equation

Which could then be solved by first-order methods.
(a) Differentiate the right-hand side of (19) and set the coefficients of y, y', and y" equa1 to find other to find g(t).
(b) Show that the integrating factor μ(t) satisfies the second-order homogeneous equation
u" - u' + u = 0
Called the adjoint equation of (18). 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(t), which might be every bit as hard as the original equation. (In Sections 4.4 and 4.5, we will develop other methods.)
(c) Show that the adjoint equation of the general second order linear equation
y" + p(t)y' + q(t)y = f(t)
Is the homogeneous equation
μ" - p(t)μ' + [q (t) - p'(t))μ = 0.

u(1)y'+g(1)y = H(1)f(1)dt +c.