EXERCISES 4.1
4.1.1 Initial-Value and Boundary-Value Problems
In Problems 1–4 the given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
1. y = c1 e^x + c2 e^{-x}, (-∞, ∞);
y'' - y = 0, y(0) = 0, y'(0) = 1
2. y = c1 e^{4x} + c2 e^{-x}, (-∞, ∞);
y'' - 3y' - 4y = 0, y(0) = 1, y'(0) = 2
3. y = c1 x + c2 x ln x, (0, ∞);
x^2 y'' - x y' + y = 0, y(1) = 3, y'(1) = -1
4. y = c1 + c2 cos x + c3 sin x, (-∞, ∞);
y''' + y' = 0, y(π) = 0, y'(π) = 2, y''(π) = -1