Question

Please show all work and steps! Would like to learn how!

Given a second order linear homogeneous differential equation a2(x)y" + a1(x)y' + 20 (x)y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions Yı, Y2. But there are times when only one function, call it Yı, 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 a2(x) #0 we rewrite the equation as 11 (1) y" +p(x)y' + g(x)y = 0 p(x) = 476) g(x) = 42(X) Then the method of reduction of order gives a second linearly independent solution as ce-/ p(x)dx ¥2 (x) = Cyı (x) / 26-dx where C is an arbitrary constant. We can choose the arbitrary constant to be anything we like. Once useful choice is to choose C so that all the constants in front reduce to 1. For example, if we obtain y2 = C3e-* then we can choose C = 1/3 so that y2 = e-x. Given the problem x?y" + 4xy' – 18y = 0 and a solution yı = x Applying the reduction of order method to this problem we obtain the following yi(x) = x^4 p(x) = and e ſux)dx = So we have re-p(x)dx (x) dx = // dx = 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) = So the general solution to 25y" – 4y' + 4y = 0 can be written as y = Ciyi + C2y2 = C1 +C2

Answer 1

Given fowblem is xy" +42y'-189=0. y't deny = 0 Or, y"+ y = 23. pix) = (1/2 and e Skesde usted -4 logee se e e Sants - Yah = 1 / X so we have le spada dx = 1 2 4 4 dx = S to da Choose a=-9 their Öz (2) = +9). to - Ya? to the general solution to 258"

254"-4y/+4y=0 Its characteristic equation is. 25m2-4m+ 4 = 0 me 4 /16-400 - 4/-384 T 50 50 4+856 2 25 = 50 0,- en moderne je ani de nostra higiene je gebou en 1949)x + ca este in 95