Differential Equations Challenge

1. The solution of y" + y' - 2y = 0 is: a) ϕ(x) = C1e-2x + C2ex b) ϕ(x) = (C1 + C2) c) ϕ(x) = C1ex + C2e-x d) ϕ(x) = C1

• ϕ(x) = C1e^x + C2e^{-2x}

• ϕ(x) = C1e^{-x} + C2e^{2x}

• ϕ(x) = C1e^{3x} + C2e^{-3x}

• ϕ(x) = C1 + C2e^{x}

2. The solution of y" + 16y = 0 is: a) ϕ(x) = (C1 + C2)e2x b) ϕ(x) = C1ei4x + C2e-i4x c) ϕ(x) = (C1 + C2)e5x d) None

• ϕ(x) = C1sin(4x) + C2cos(4x)

• b) ϕ(x) = C1ei4x + C2e-i4x

• ϕ(x) = (C1 + C2)e3x

• ϕ(x) = C1e2x + C2e-2x

3. If ψ(x) = C1e2x + C2e-3x, find ψ(0) = 0, ψ'(0) = ?

• 5C1

• 5C2

• -3C2

• C1e2

4. If ϕ1(x) = x, ϕ2(x) = 1, are they: a) Linearly Independent b) Linearly Dependent c) Independent d) Dependent

• d) Unrelated

• c) Non-orthogonal

• a) Orthogonal

• b) Linearly Dependent

5. If ϕ1 and ϕ2 are two solutions of L(y) = 0, then if ϕ1, ϕ2 are Linearly Dependent, w(ϕ1, ϕ2) = ? a) 0 b) 1 c) 2 d) 4

• -1

• 3

• 0

• 5

6. What is the general solution of y" + 4y = 0? a) ϕ(x) = C1cos(2x) + C2sin(2x) b) ϕ(x) = C1e2x + C2e-2x c) ϕ(x) = C1e4x + C2e-4x d) None

• a) ϕ(x) = C1cos(2x) + C2sin(2x)

• ϕ(x) = C1e3x + C2e-3x

• ϕ(x) = C1cos(4x) + C2sin(4x)

• ϕ(x) = C1cos(x) + C2sin(x)

7. The characteristic equation of y" - 5y' + 6y = 0 is: a) r^2 - 5r + 6 = 0 b) r^2 + 5r + 6 = 0 c) r^2 - 6r + 5 = 0 d) None

• a) r^2 - 5r + 6 = 0

• r^2 - 4r + 6 = 0

• r^2 - 5r - 6 = 0

• r^2 + 6r + 5 = 0

8. If ϕ(x) = e^(-x), what is ϕ'(x)? a) e^(-x) b) -e^(-x) c) 0 d) None

• -x

• 1

• -e^(-x)

• e^(x)

9. The Wronskian of two functions ϕ1 and ϕ2 is defined as: a) ϕ1ϕ2' - ϕ2ϕ1' b) ϕ1'ϕ2 - ϕ2'ϕ1 c) ϕ1 + ϕ2 d) None

• ϕ1 - ϕ2'

• ϕ1' + ϕ2'

• b) ϕ1'ϕ2 - ϕ2'ϕ1

• ϕ1ϕ2 + ϕ1'ϕ2'

10. The solution of y" + 9y = 0 is: a) ϕ(x) = C1cos(3x) + C2sin(3x) b) ϕ(x) = C1e3x + C2e-3x c) ϕ(x) = C1e9x + C2e-9x d) None

• ϕ(x) = C1cos(3x) + C2e3x

• ϕ(x) = C1e3x + C2e-9x

• a) ϕ(x) = C1cos(3x) + C2sin(3x)

• ϕ(x) = C1cos(9x) + C2sin(9x)