Ordinary Differential Equations
Review
Higher-Order DEs
18 problems
- Solve y″ − 3y′ = 8e^{3x} + 4 sin x using an annihilator and undetermined coefficients
- Solve y″ + 3y′ + 2y = 4x² using an annihilator and undetermined coefficients
- Solve y″ + y = x cos x − cos x using an annihilator and undetermined coefficients
- Solve the Cauchy–Euler equation 4x²y″ + 8xy′ + y = 0
- Solve the Cauchy–Euler equation x²y″ − 2xy′ − 4y = 0
- Solve the system x′ = 2x − y, y′ = x by elimination
- Solve the system x′ + y′ = −2y, x′ = 3x + 2y by elimination
- Solve the system x′ − 4x + y″ = t², x′ + x + y′ = 0 by elimination
- Solve the system x′ = −y + t, y′ = x − t by elimination
- Use a Green's function to solve y″ + 4y = sin 2x with y(0) = 1, y′(0) = −2
- Use a Green's function to solve y″ + 4y = x with y(0) = 0, y′(0) = 0
- Solve 4y″ − y = xe^{x/2} with y(0) = 1, y′(0) = 0 by variation of parameters
- Derive a particular solution of y′ + P(x)y = g(x) by variation of parameters, given a solution of the homogeneous equation
- Derive the variation-of-parameters formula for a particular solution of y″ + P(x)y′ + Q(x)y = g(x)
- Solve y″ − 4y′ + 4y = (x + 1)e^{2x} by variation of parameters
- Solve y″ + 2y′ − 8y = 2e^{−2x} − e^{−x} with y(0) = 1, y′(0) = 0 by variation of parameters
- Solve y″ + y = sec x by variation of parameters
- Solve y″ + y = sin x by variation of parameters
Laplace Transform
25 problems
- Derive ℒ{e^{at} f(t)} = F(s − a)
- Solve the integral equation f(t) = 1 + t − (8/3) ∫₀^{t} (τ − t)³ f(τ) dτ
- Solve the integral equation f(t) + ∫₀^{t} f(τ) dτ = 1
- Solve the integral equation f(t) + ∫₀^{t} (t − τ) f(τ) dτ = t
- Solve the integral equation f(t) = te^{t} + ∫₀^{t} τ f(t − τ) dτ
- Evaluate ℒ⁻¹{(2s + 5)/(s − 3)²} by partial fractions
- Solve x″ + 16x = cos 4t with x(0) = 0, x′(0) = 1
- Solve y″ − 3y′ + 2y = e^{−4t} with y(0) = 1, y′(0) = 5
- Solve y″ − 6y′ + 13y = 0 with y(0) = 0, y′(0) = −3
- Solve y″ − 6y′ + 9y = t with y(0) = 0, y′(0) = 1
- Solve y″ − 6y′ + 9y = t²e^{3t} with y(0) = 2, y′(0) = 17
- Solve y″ − y′ = e^{t} cos t with y(0) = 0, y′(0) = 0
- Solve y″ + 2y′ + y = 0 with y(0) = 1, y′(0) = 1
- Solve y″ + 4y = 𝒰(t − 2π) sin t with y(0) = 1, y′(0) = 0
- Solve y″ + y = δ(t − 2π) with y(0) = 0, y′(0) = 1
- Solve y″ + y = f(t), y(0) = 0, y′(0) = 1, where f(t) = 1 for π ≤ t < 2π and 0 otherwise
- Solve y″ + y = δ(t − π/2) + δ(t − 3π/2) with y(0) = 0, y′(0) = 0
- Solve y′ − 3y = δ(t − 2) with y(0) = 0
- Solve y′ − y = 1 with y(0) = 0
- Solve y′ + 2y = f(t), y(0) = 0, where f(t) = t for 0 ≤ t < 1 and 0 for t ≥ 1
- Solve y′ + 3y = 13 sin 2t with y(0) = 6
- Solve y′ + 4y = e^{−4t} with y(0) = 2
- Solve y′ + 6y = e^{4t} with y(0) = 2
- Solve y′ + y = f(t), y(0) = 0, where f(t) = 0 for 0 ≤ t < 1 and 5 for t ≥ 1
- Solve y′ + y = t sin t with y(0) = 0