Question

HW3.2: Problem 1 Previous Problem Problem List Next Problem (1 point) Given a second order linear homogeneous differential equation a2(x)y" + ai (x)y' + ao (x)y0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yi, y2. But there are times when only one function, call it y, is available and we would like to find a second linearly independent solution. We can find 2 using the method of reduction of order. First, under the necessary assumption the a2(x)0 we rewrite the equation as ai (x) a2(x) ao (x) a2(x) Then the method of reduction of order gives a second linearly independent solution as

Answer 1

9 y'' - 12 y' + 4y = 0 y_1 = e^2x/3 therefore y^2_1(x) = (e^2x/3)^2 = e^2(2x/3) = e^4x/3 therefore y^2_1(x) = e^(4x/3) = e^(4x/3) 9y'' - 12y' + 4y = 0 implies y'' - 12/9 y' + 4/9 y = 0 p(x) = -12/9 = -4/3 p(x) = -4/3 = -4/3 e^-integral p(x) dx = e^- integral - 4/3 dx = e^(4/3 x) e^- integral p(x) dx = e^(4/3 x) integral e^- integral p(x) dx/y^2_1(x) dx = integral e^(4/3 x)/e^(4/3 x) dx = integral dx = x therefore integral e^- integral p(x) dx/y^2_1 dx dx = integral e^(4/3 x)/e^(4/3 x) dx = x y_2(x) = c y_1 (x) integral e^- integral p(x) dx/y^2_1 (x) dx y_2(x) = c e^2x/3 x = c x e^2x/3 y_2(x) = x e^(2x/3) y = c_1 y_1 + c_2 y_2 = c_1 e^(2x/3) + c_2 x e^(2x/3) y = c_1 y_1 + c_2 y_2 = c_1 e^(2x/3) + c_2 x e^(2x/3)