Quiz in MATH 223 - Part 3

Find the orthogonal trajectories of the given family of curves x² + y² = C².

• y = kx
• xy = C
• y² = kx
• y² = kxy

What is the general solution to (2xy – 3y) dx + (x – 3x) dy = 0 ?

• xy (x – 3) = C
• x (xy – 4) = C
• xy + 2x² = C
• 2xy + 3x² = C

Eliminate the arbitrary constants: y = c₁e²ˣ + c₂e⁻⁴ˣ

• y" - 2y' - 8y = 0
• y" + 8y' - 2y = 0
• y" + 2y' - 8y = 0
• y" - 8y' - 2y = 0

Eliminate the arbitrary constants: y = (c₁ + c₂x)e²ˣ

• y" + 4y' - 4y = 0
• y" + 4y' + 4y = 0
• y" - 4y' - 4y = 0
• y" - 4y' + 4y = 0

Eliminate the arbitrary constants: y = c₁e²ˣ + c₂e⁴ˣ + c₃e⁵ˣ

• y''' - 11y" + 38y' - 40y = 0
• y''' - 11y" - 38y' + 40y = 0
• y''' + 11y" - 38y' - 40y = 0
• y''' - 11y" - 38y' - 40y = 0

Eliminate the arbitrary constants: y = e²ˣ(c₁cos3x + c₂sin3x)

• y" - 4y' - 13y = 0
• y" + 4y' - 13y = 0
• y" - 4y' + 13y = 0
• y" + 4y' + 13y = 0

∫e²ˣcos9x dx

• y = [(9/85)sin9x + (2/85)cos9x] + Ce⁻²ˣ

• y = [(2/85)sin9x - (9/85)cos9x] + Ce⁻²ˣ

• y = [(2/85)cos9x - (9/85)sin9x] + Ce⁻²ˣ

• y = [-(2/85)sin9x - (9/85)cos9x] + Ce⁻²ˣ

∫e²ˣsin9x dx

• y = [(9/85)sin9x + (2/85)cos9x] + Ce⁻²ˣ

• y = [(2/85)sin9x - (9/85)cos9x] + Ce⁻²ˣ

• y = [(2/85)cos9x - (9/85)sin9x] + Ce⁻²ˣ

• y = [-(2/85)sin9x - (9/85)cos9x] + Ce⁻²ˣ

What is the solution to the following differential equation y' + 5y = 0?

• A. y = 5x + C
• B. y = Ce⁻⁵ˣ
• C. y = Ce⁵ˣ
• D. either A or B

What is the general solution of the given differential equation dy/dx – y/x = 2 ln x

• y = x ln²x + Cx
• y = x lnx + Cx²
• y = 4x lnx + C
• y = 4x lnx + Cx²