Differential Equations Third Exam Spring 2024

Exam Structure

• Topics Covered: Thea transforms, series
• Number of Problems: 6
• Initial Step: Write your name on the exam

Problem 1: Inverse Laplace Transform

• Given: (\frac{2s^3 - 1}{s^2 (s^2 + 1)})
• Task: Find the inverse Laplace transform
• Procedure:
  • Partial Fractions: Break down the fraction using constants A, B, C, D.
  • Identify Constants: Use s = 0 to find B, then use polynomial comparison.
  • Results:
    • ( B = -1 )
    • ( A = 0 )
    • ( C = 2 )
    • ( D = 1 )
  • Laplace Components: (-1 \cdot \frac{1}{s^2}, \, 2s \pm 1\cdot\frac{1}{s^2 + 1})
  • Inverse Laplace: (-t + 2\cos(t) + \sin(t))

Problem 2: Laplace Transform of Given Functions

• Given: (\frac{s}{s^4 + 1} + F(0) = 0, F'(0) = 0)
• Task: Identify the Laplace transforms for a set of given equations
• Methods:
  • Using Rules: Apply rules for derivatives and convolutions
• Results:
  • (\mathcal{L}(f'(t)) = \frac{s^3}{s^4 + 1})
  • (\mathcal{L}(tf(t)) = \frac{3s^4 - 1}{s^4 + 1})
  • (\mathcal{L}(\int_0^t f(\tau)f(t-\tau)d\tau) = \frac{s^2}{s^4 + 1}^2)
  • (\mathcal{L}(e^{5t}f(t)) = \frac{s-5}{(s-5)^4 + 1})
  • (\mathcal{L}(u(t-5)f(t-5)) = e^{-5s} \mathcal{L}(f(t)))

Problem 3: Piecewise Function Laplace Transform

• Given: (f(t)): $4t$ if $t<2$ and $t^2 + 4$ if $t \geq 2$
• Part A Task: Find the Laplace transform
• Method:
  • Rewrite the function by using Heaviside function
  • Transform Parts: (\mathcal{L}(4t), \mathcal{L}(U(t-2)[t^2 + 4 - 4t]))
  • Results:(\frac{4}{s^2} + \frac{2e^{-2s}}{s^3})

Problem 3: Part B

• Given: Integrable function in the complex form
• Express: Integrand as a convolution
• Result: Transform using gamma function
  • Result: (\mathcal{L}(F * t^{-1/2}) = \frac{4\sqrt{\pi}}{s^{2.5}})

Problem 4: Mass-Spring System

• Given: Mass, spring constant, initial conditions
• Part A Task: Set up the differential equation
• Result: (X'' + 4X = 2\delta(t-3) + 7\delta(t-5))
• Part B Task: Solve the differential equation using Laplace
  • Equation: (\mathcal{L}(X'') + 4\mathcal{L}(X) = 2e^{-3s} + 7e^{-5s})
  • Solution: (X(t) = 2U(t-3)\sin(2(t-3)) + 7U(t-5)\sin(2(t-5)))

Problem 5: Series Solution to Differential Equation

• Given: (Y = \sum_{n=0}^{\infty}a_nx^n)
• Task: Find recurrence relation
• Key Steps:
  • Identify derivatives
  • Substitute series into differential equation
  • Collect terms/powers
• Results: (a_{n+2} = \frac{a_n(-2n-1)}{(n+2)(n+1)})

Problem 6: Finding Coefficients in Series

• Given: Series for Y and a differential equation
• Initial Conditions: Given explicitly
• Task: Find B0, B1, B2, B3
  • Set up equations using initial conditions
  • Substitute and solve equations
• Results:
  • (B_0 = 1)
  • (B_1 = 0)
  • (B_2 = 1)
  • (B_3 = -\frac{1}{6})

Summary

• Follow systematic approaches in solving differential equations using the Laplace transform
• Understand the application of partial fractions and convolution technique
• Utilize initial conditions effectively in series solutions