Final Exam in MATH 223 - Part 3

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

• y = kx
• x² + y² = ky
• y² = kx
• y² = kxy

Find the orthogonal trajectories of the circles x² + y² = cx ?

• y = kx
• x² + y² = ky
• y² = kx
• y² = kxy

A bacterial population is known to have a rate of growth proportional to itself. If between noon and 2 P.M., the population triples, at what time, no controls being exerted, should the population becomes 100 times as what it was at noon?

• 8:23 PM
• 9:35 PM
• 9:35 PM
• 10:37 PM

At 9 A.M., a thermometer reading 70°F is taken outside where the temperature is 15°F. At 9:05 A.M., the thermometer reading is at 45°F. At 9:10 A.M., the thermometer is taken back indoors where the temperature is fixed at 70°F, find the reading at 9:15 A.M.

At 2:00 P.M. a thermometer reading 80 °F is taken outside where the air temperature is 20 °F. At 2:05 P.M., the temperature reading yielded by the thermometer is 40 °F. Later the thermometer is brought inside where the air is at 80 °F. At 2:10 P.M. the reading is 70 °F. When was the thermometer brought indoors?

Solve: d²y/dx² - 7dy/dx + 12 = 0

• y = C₁e³ˣ + C₂e⁻⁴ˣ
• y = C₁e⁻³ˣ + C₂e⁴ˣ
• y = C₁e³ˣ + C₂e⁴ˣ
• y = C₁e⁻³ˣ + C₂e⁻⁴ˣ

Solve: d²y/dx² - 4dy/dx + 4y = 0

• y = (C₁ + C₂x) e⁴ˣ
• y = (C₁ + C₂x) e²ˣ
• y = C₁e²ˣ + C₂e²ˣ
• y = (C₁ + C₂x) e⁻²ˣ

Solve: d²y/dx² - 8dy/dx + 25y = 0

• y = e⁴ˣ (C₁ cos3ix + C₂ sin3ix)
• y = e³ˣ (C₁ cos4x + C₂ sin4x)
• y = e⁻⁴ˣ (C₁ cos3x + C₂ sin3x)
• y = e⁴ˣ (C₁ cos3x + C₂ sin3x)

Solve: [d²y/dx² + 36]² = 0

• y = (C₁ + C₂x) cos6x + (C₃ + C₄x) sin6x
• y = (C₁ + C₂x) cos6ix + (C₃ + C₄x) sin6ix
• y = (C₁ + C₂) cos6x + (C₃ + C₄x) sin6x
• y = (C₁ + C₂x) cos6x + (C₃ + C₄) sin6x

Solve the given Non-Homogeneous Differential equation: (D² + 4D + 3)y = 5e²ˣ

• y = c₁e³ˣ + c₂e⁻ˣ + e²ˣ/3
• y = c₁e⁻³ˣ + c₂e⁻ˣ + e²ˣ/3
• y = c₁e⁻³ˣ + c₂eˣ + e²ˣ/3
• y = c₁e³ˣ + c₂eˣ + e²ˣ/3