Which of the following statements are true? Do not try to solve the differential equations explicitly:
(a) Consider the differential equation
y' = 1/(1+x^4) y + cos(3x)
Depending on the initial condition y(0), the solution may not exist for all values of x (blow up in finite x).
(b) If y(x) is a solution of the differential equation
y' = (1 - x)y^2,
then for any real number c, cy(x) is also a solution of the same equation.
(c) If y1(x) and y2(x) are solutions of the differential equation
y'' + e^(-5x)y' + sin(2x)y = ln(1 + x^2),
then y1(x) + y2(x) is also a solution of the same equation.
(d) The functions 1/(1+x^4), (1 - x), and sin(2x) are linearly independent.