GELE Mathematics — Differential EquationsRevision Notes

Formulas

Exam Tips

• PRC multiple-choice items often ask: 'What is the order and degree of the following DE?' — practice identifying these rapidly.
• If derivatives appear inside a radical or as a denominator, always clear them algebraically before classifying.
• Linear vs. nonlinear: scan the equation for y², (y')², y·y', sin(y), e^y — any of these signals nonlinearity.

Key Points

• Order is the highest derivative present in the equation. Degree is the power of the highest-order derivative after the equation is made polynomial in its derivatives (radicals and fractions in derivatives must be cleared first).
• An ordinary differential equation (ODE) involves derivatives with respect to a single independent variable. A partial differential equation (PDE) involves partial derivatives with respect to two or more independent variables.
• A DE is linear if the dependent variable and all its derivatives appear to the first power only, with coefficients that are functions of the independent variable alone. Any product of the unknown function with its derivative, or any nonlinear function of the unknown (e.g., sin y, y²), makes it nonlinear.
• For the PRC exam, focus on ODEs — first-order and second-order linear with constant coefficients.
• The general solution contains as many arbitrary constants as the order. A particular solution applies initial or boundary conditions to fix those constants.

Definitions

Section Title

1. Classification of Differential Equations

Common Mistakes

• Confusing order and degree: y'' + (y')³ = 0 is 2nd-order (highest derivative is y'') but 1st-degree in y'' (y'' appears to the first power after the equation is written as a polynomial in y'').
• Forgetting to clear radicals before stating the degree: √(y'') = x is rewritten as y'' = x² (degree 1), not degree 1/2.
• Calling a 2nd-order equation 'degree 2' simply because it is second-order — order and degree are different properties.
• Claiming an ODE is nonlinear because the coefficients contain x (e.g., xy' + y = 0 is still linear — nonlinearity refers to y and its derivatives, not x).

Formulas

Exam Tips

• Decision rule: (1) Can you separate all y's from all x's? → Separable. (2) Is it dy/dx + P(x)y = Q(x)? → Integrating factor. (3) Does ∂M/∂y = ∂N/∂x? → Exact.
• Always present the answer as y = f(x) unless instructed otherwise; board problems nearly always want the explicit form.
• When initial conditions are given (e.g., y(0) = 1), solve for C immediately after integrating — do not carry C to the final answer.
• For exam speed: memorize that μ = e^(∫P dx) and d(μy)/dx = μQ; avoid re-deriving this under time pressure.

Key Points

• Three main solution methods for first-order ODEs on the PRC exam: (1) Separation of Variables, (2) Linear First-Order (Integrating Factor), (3) Exact Equations.
• Always identify the type first before choosing the solution method — this is the most important skill.
• Separable: all y-terms (including dy) on one side, all x-terms (including dx) on the other, then integrate both sides.
• Linear first-order: standard form is dy/dx + P(x)y = Q(x). The integrating factor μ = e^(∫P dx) converts the left side into d(μy)/dx.
• Exact: M dx + N dy = 0 is exact when ∂M/∂y = ∂N/∂x. Solution found by integrating M with respect to x (or N with respect to y), then adjusting for the remaining variable.
• Bernoulli equation dy/dx + P(x)y = Q(x)yⁿ is nonlinear but is reduced to a linear DE by the substitution v = y^(1-n).

Definitions

Section Title

2. First-Order Differential Equations

Common Mistakes

• Forgetting the constant of integration when integrating both sides of a separable equation — this constant is essential for applying the initial condition.
• In the integrating factor method, applying μ only to one side: multiply BOTH sides (left side and Q(x)) by μ.
• Errors in ∫P dx for the integrating factor when P involves absolute values or requires substitution.
• For exact equations: integrating M w.r.t. x and then forgetting that the 'constant' may actually be a function of y; always differentiate F w.r.t. y and equate to N to find that function.
• Dividing by h(y) when separating variables without checking if h(y) = 0 — this can lose singular solutions.

Formulas

Exam Tips

• Memorize the three root cases and their solution forms — this is tested almost every board exam cycle.
• For 'solve with initial conditions y(0)=a, y'(0)=b': write the general solution first, then differentiate, then apply both conditions simultaneously.
• Discriminant shortcut: b² - 4ac > 0 → distinct real; = 0 → repeated; < 0 → complex conjugate.
• The method of undetermined coefficients guess table: f(x) = polynomial of degree n → guess Aₙxⁿ + ... + A₀; f(x) = e^(kx) → guess Ae^(kx); f(x) = sin(kx) or cos(kx) → guess A·sin(kx) + B·cos(kx).

Key Points

• For a homogeneous linear ODE with constant coefficients (right side = 0), the solution method is: assume y = e^(mx), substitute, and solve the resulting characteristic (auxiliary) equation.
• The nature of the roots of the characteristic equation completely determines the form of the general solution.
• Three root cases: (a) distinct real roots, (b) repeated real roots, (c) complex conjugate roots — each gives a different solution structure.
• For non-homogeneous equations (right side ≠ 0), the general solution = complementary function (yc) + particular integral (yp). The complementary function is the homogeneous solution; the particular integral is found by undetermined coefficients or variation of parameters.
• The most common second-order case on the PRC exam: ay'' + by' + cy = f(x).
• Undetermined coefficients: guess yp based on the form of f(x) — polynomial, exponential, sine/cosine, or combinations.

Definitions

Section Title

3. Higher-Order Linear ODEs with Constant Coefficients

Common Mistakes

• Using y = C₁e^(m₁x) + C₂e^(m₂x) when roots are repeated — the correct form is (C₁ + C₂x)e^(mx).
• For complex roots α ± βi, writing y = C₁e^(αx)cos(βx) + C₂e^(αx)sin(βx) is correct, but students often forget the e^(αx) factor when α ≠ 0.
• When f(x) = Ae^(mx) and m is already a root of the characteristic equation, the usual guess for yp must be multiplied by x (resonance case).
• Sign errors when computing the discriminant b² - 4ac: a negative discriminant means complex roots with β = √(4ac - b²)/(2a).
• Forgetting to find C₁ and C₂ using initial conditions when a particular solution is requested.
• Dropping the particular integral and presenting only the complementary function as the final answer.

Formulas

Exam Tips

• For growth/decay board problems: (1) Write y = y₀e^(kt); (2) Use the given data point to find k; (3) Substitute the required time to find the answer.
• For Newton's cooling: rewrite as dT/dt + k·T = k·Ts (linear form) or solve directly as separable. Both work — use whichever is faster for you.
• Mixing tank problems almost always result in a linear first-order ODE — apply the integrating factor method directly.
• Always verify the sign of k against the physical situation: cooling means T decreases → k must be positive in the formula T = Ts + (T₀-Ts)e^(-kt).
• Doubling time Td = ln(2)/k is frequently asked; memorize this alongside the half-life formula.

Key Points

• Growth and Decay: dy/dt = ky. k > 0 → growth (population, compound interest); k < 0 → decay (radioactivity, concrete carbonation). Solution: y = y₀e^(kt).
• Half-life T₁/₂ = ln(2)/|k|. Doubling time Td = ln(2)/k.
• Newton's Law of Cooling: dT/dt = -k(T - Ts), where Ts is the surrounding temperature and k > 0. Solution: T - Ts = (T₀ - Ts)e^(-kt). Used in forensic engineering and material temperature monitoring.
• Mixing Problems (Tank Problems): d(Q)/dt = (rate in) - (rate out). Set up the ODE from mass balance; Q is the amount of substance (e.g., salt in kg) in the tank at time t.
• Simple Harmonic Motion (undamped): d²x/dt² + ω²x = 0 → x = C₁cos(ωt) + C₂sin(ωt), where ω = √(k/m) for a spring-mass system.
• Laplace Transform: converts a linear ODE with initial conditions into an algebraic equation in the s-domain; the solution is found algebraically and inverted back. Powerful for systems with discontinuous or impulsive forcing.

Definitions

Section Title

4. Engineering Applications of Differential Equations

Common Mistakes

• Growth/Decay: forgetting that decay requires k < 0; using a positive k and then subtracting — be consistent with the sign convention in the ODE setup.
• Newton's cooling: writing dT/dt = k(T - Ts) with k > 0 (wrong sign) leading to an unbounded (growing) temperature — physically impossible for cooling. Always use dT/dt = -k(T - Ts) with k > 0.
• Mixing problems: using the wrong volume V when inflow rate ≠ outflow rate (the volume is not constant in that case — V must be expressed as a function of t).
• Forgetting to convert units consistently — e.g., mixing rate in L/min vs. L/hr, or half-life in years vs. seconds.
• In the mixing tank, confusing Q (total amount in kg) with concentration (kg/L); the outflow concentration is Q/V, not Q.

Formulas

Exam Tips

• Maintain a compact Laplace transform table on your reference sheet for the board exam.
• When doing partial fractions, verify by recombining — this is a quick self-check under exam conditions.
• For PRC exam: Laplace problems usually involve at most 2nd-order ODEs with polynomial/exponential right-hand sides — master the standard five transform pairs and derivative rules.

Key Points

• The Laplace transform converts a linear ODE with initial conditions directly into an algebraic equation in the complex frequency domain (s-domain), eliminating the need to find the complementary function and particular integral separately.
• Definition: L{f(t)} = F(s) = ∫₀^∞ e^(-st)f(t) dt.
• Key transforms to memorize: L{1} = 1/s; L{t^n} = n!/s^(n+1); L{e^(at)} = 1/(s-a); L{sin(bt)} = b/(s²+b²); L{cos(bt)} = s/(s²+b²).
• Transform of derivatives: L{f'(t)} = sF(s) - f(0); L{f''(t)} = s²F(s) - sf(0) - f'(0). This is how initial conditions are automatically incorporated.
• After transforming to the s-domain, solve the algebraic equation for F(s), then use inverse Laplace transform (partial fractions) to recover y(t).
• On the PRC exam, Laplace problems are typically straightforward: transform a simple ODE, solve for F(s), and invert using standard table entries.

Definitions

Section Title

5. Laplace Transform Method (Introduction)

Common Mistakes

• Forgetting to include initial condition terms when transforming derivatives — L{y'} = sY - y(0), not just sY.
• Errors in partial fraction decomposition when inverting F(s) — practice this algebra step thoroughly.
• Applying Laplace transform to nonlinear ODEs (it only works for linear ODEs).
• Confusing the transform pairs: L{sin(bt)} = b/(s²+b²) vs L{cos(bt)} = s/(s²+b²) — note the numerator differs.