(1 point) Given a second order linear homogeneous differential equation a2(x)y\" + a1(x);/ + ao(x)y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yl , y2. But there are times when only one function, call it yl , is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the (12(x) # 0 we rewrite the equation as y\" + p(X)J/ + q(X)y = 0 p(X) = Then the method of reduction of order gives a second linearly independent solution as e fp(x)dx y2(X) = Cylu = CytOC) 711x where C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. One useful choice is to choose C so that all the constants in front reduce to 1. For example, if we obtain y2 = C3e2" then we can choose C = 1/3 so that y2 = ez". Given the problem xzy\" +4x/ 18y = 0 and a solution y] = x3 Applying the reduction of order method to this problem we obtain the following y?(X) = p(x) = and [\"00\" = So we have e /p(x)dx d / d x = x = nyc) Finally, after making a selection of a value for C as described above (you have to choose some nonzero numerical value) we arrive at y2(X) = Cylu = So the general solution to 25y\" 4y' + 4y = 0 can be written as y = 61311 + CzYz = C] +02