Differential Equations
Key Takeaways
- First-order ODEs: separable, linear (integrating factor), and exact equations are the main types tested.
- Second-order linear ODEs with constant coefficients use the characteristic equation ar² + br + c = 0.
- The characteristic equation roots determine the solution form: real distinct, repeated, or complex conjugate.
- Laplace transforms convert differential equations to algebraic equations, simplifying the solution process.
- Initial value problems (IVPs) require specific initial conditions to find the particular solution.
- The FE Reference Handbook contains Laplace transform pairs — know how to use the table.
Differential Equations
Differential equations describe how quantities change and are fundamental to modeling engineering systems — from circuit analysis to heat transfer to vibrations.
First-Order Ordinary Differential Equations (ODEs)
Separable Equations
An ODE is separable if it can be written as:
Solution method: Separate variables and integrate both sides:
Example: dy/dx = xy → ∫dy/y = ∫x dx → ln|y| = x²/2 + C → y = Aeˣ²/²
First-Order Linear Equations
Standard form: dy/dx + P(x)y = Q(x)
Solution using integrating factor μ(x):
Example: dy/dx + 2y = 6
- P(x) = 2, so μ = e²ˣ
- y = e⁻²ˣ ∫ 6e²ˣ dx = e⁻²ˣ (3e²ˣ + C) = 3 + Ce⁻²ˣ
Exact Equations
An equation M(x,y)dx + N(x,y)dy = 0 is exact if ∂M/∂y = ∂N/∂x.
Solution: Find F(x,y) such that ∂F/∂x = M and ∂F/∂y = N.
Second-Order Linear ODEs with Constant Coefficients
The general form:
Homogeneous Case (f(x) = 0)
Solve the characteristic equation: ar² + br + c = 0
| Root Type | Condition | General Solution |
|---|---|---|
| Real, distinct (r₁ ≠ r₂) | b² - 4ac > 0 | y = C₁eʳ¹ˣ + C₂eʳ²ˣ |
| Real, repeated (r₁ = r₂ = r) | b² - 4ac = 0 | y = (C₁ + C₂x)eʳˣ |
| Complex conjugate (α ± βi) | b² - 4ac < 0 | y = eᵅˣ(C₁ cos βx + C₂ sin βx) |
Non-Homogeneous Case (f(x) ≠ 0)
General solution = Homogeneous solution (yₕ) + Particular solution (yₚ)
Method of Undetermined Coefficients — guess yₚ based on f(x):
| f(x) | Guess for yₚ |
|---|---|
| Constant k | A |
| Polynomial (degree n) | Aₙxⁿ + Aₙ₋₁xⁿ⁻¹ + ... + A₀ |
| eᵅˣ | Aeᵅˣ |
| sin(βx) or cos(βx) | A cos(βx) + B sin(βx) |
| eᵅˣ sin(βx) | eᵅˣ(A cos βx + B sin βx) |
Important: If your guess duplicates a term in yₕ, multiply by x (or x² if needed) to ensure independence.
Laplace Transforms
The Laplace transform converts a function of time f(t) to a function of complex frequency F(s):
Common Laplace Transform Pairs
| f(t) | F(s) |
|---|---|
| 1 | 1/s |
| t | 1/s² |
| tⁿ | n!/sⁿ⁺¹ |
| eᵅᵗ | 1/(s-a) |
| sin(ωt) | ω/(s²+ω²) |
| cos(ωt) | s/(s²+ω²) |
| eᵅᵗ sin(ωt) | ω/((s-a)²+ω²) |
| eᵅᵗ cos(ωt) | (s-a)/((s-a)²+ω²) |
Key Properties
| Property | Time Domain | s-Domain |
|---|---|---|
| Linearity | af(t) + bg(t) | aF(s) + bG(s) |
| First Derivative | f'(t) | sF(s) - f(0) |
| Second Derivative | f''(t) | s²F(s) - sf(0) - f'(0) |
| Time Shift | f(t-a)u(t-a) | e⁻ᵃˢF(s) |
| s-Shift | eᵅᵗf(t) | F(s-a) |
Solving ODEs with Laplace Transforms
- Take the Laplace transform of both sides of the ODE
- Substitute initial conditions
- Solve algebraically for Y(s)
- Take the inverse Laplace transform to find y(t)
Example: Solve y'' + 4y = 0 with y(0) = 1, y'(0) = 0
Step 1: s²Y(s) - s(1) - 0 + 4Y(s) = 0 Step 2: Y(s)(s² + 4) = s Step 3: Y(s) = s/(s² + 4) Step 4: y(t) = cos(2t)
What is the general solution of y'' - 5y' + 6y = 0?
What is the Laplace transform of f(t) = e³ᵗ?
Solve the separable ODE dy/dx = 3x²y with y(0) = 2.