Use the method of reduction of order to find a second solution
y2 of the given differential equation such that {y1, y2} is a
fundamental set of solutions on the given interval.
t2y′′ +2ty′ −2y=0, t > 0, y1(t)=t
(a) Verify that the two solutions that you have obtained are
linearly independent.
(b) Let y(1) = y0, y′(1) = v0. Solve the initial value problem.
What is the longest interval on which the initial value problem is
certain to have...
Problem 3: Find a second solution by reduction of order -
nonhomogeneous
The given function y1(x) is a solution of the associated
homogeneous equation. Use the reduction of order method to find a
solution y(x) = u(x)y1(x) of the nonhomogeneous equation.
1. x^2y'' + xy'−4y = x^3, y1 = x^2
2. 2x^2*y'' + 3xy'−y = 1/x , y1 = x^(1/2)
3. Consider4 the homogenous linear second order differential
equation
y′′ − 2y′ + y = 0 (⋆)
(a) Verify that the function y = e^x is a solution of equation
(⋆) on the interval (−∞, ∞).
(b) Verify that the function y = xex is a solution of equation
(⋆) on the interval (−∞, ∞).
(c) Verify that y = 7e^x + (5xe)^x is a solution of equation
(⋆) on the interval (−∞, ∞).
(d) Assume that c and d...