2. Consider the IVP of the second order linear ODE y + p(t)y/ + q(t)y =...

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2. Consider the IVP of the second order linear ODE y" + p(t)y/ + q(t)y = g(t), y(0) = y0, y(0) = %6- Given y1(t) +0 as one fundamental solution of the homogeneous ODE, prove that (a) (8) the other fundamental solution to the homogeneous ODE can be - Sp(t)dt -dt. yi (t) y2(t) = y1 (t) (b) (4) Show that W (41, 42) (t) = Yı (t) 2(t) (t) (t) W (31. 2) (0) exp (-/ p(s-)ds). p(s)e (c) (4) Show that a solution to the non-homogeneous ODE is given by 1 Y(t) = -y1(t) 2(s)g(s)e6p(e)duds+ y>(t) [ v1(s)g(s)eS% p(u)du ds W (41, 42)(0) (d) (4) Show that the solution to the IVP is given by y(t) = C1y1(t) + Cay2(t) + Y (t), where Y0 2(0) % (0) C1 = W (y1, 42)(0) y1(0) yo (0) y% W(y1, 42) (0) and C2 = 2. Consider the IVP of the second order linear ODE y" + p(t)y/ + q(t)y = g(t), y(0) = y0, y(0) = %6- Given y1(t) +0 as one fundamental solution of the homogeneous ODE, prove that (a) (8) the other fundamental solution to the homogeneous ODE can be - Sp(t)dt -dt. yi (t) y2(t) = y1 (t) (b) (4) Show that W (41, 42) (t) = Yı (t) 2(t) (t) (t) W (31. 2) (0) exp (-/ p(s-)ds). p(s)e (c) (4) Show that a solution to the non-homogeneous ODE is given by 1 Y(t) = -y1(t) 2(s)g(s)e6p(e)duds+ y>(t) [ v1(s)g(s)eS% p(u)du ds W (41, 42)(0) (d) (4) Show that the solution to the IVP is given by y(t) = C1y1(t) + Cay2(t) + Y (t), where Y0 2(0) % (0) C1 = W (y1, 42)(0) y1(0) yo (0) y% W(y1, 42) (0) and C2 =