Differential Equations: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

Differential equations

Exponential growth/decay: dy/dt = ky → y = y₀e^kt
Half-life: k = ln 2 / t_½
Newton's law of cooling: T(t) = T_a + (T₀ − T_a)e^−kt
Integrating factor: y′ + p(x)y = q(x): μ = e^{∫p dx}, (μy)′ = μq
Mixing tank: dm/dt = r·c − (r/V)·m

Integrals

Power rule: ∫xⁿ dx = xⁿ⁺¹/(n + 1) + C
Definite integral: ∫_a^b f dx = F(b) − F(a)
Common integrals: ∫eˣ dx = eˣ, ∫1/x dx = ln|x|, ∫sin x dx = −cos x
Area between curves: ∫_a^b (f − g) dx

Method

Rewrite the integrand as a sum of simple terms where possible.
Apply the power rule: ∫xⁿ dx = xⁿ⁺¹/(n + 1), and standard integrals for e^x, 1/x, sin and cos.
Use substitution when you see a function and its derivative together.
For a definite integral, evaluate F(b) − F(a).

Derivatives

Power rule: (xⁿ)′ = n·xⁿ⁻¹
Product rule: (uv)′ = u′v + uv′
Quotient rule: (u/v)′ = (u′v − uv′)/v²
Chain rule: f(g(x))′ = f′(g(x))·g′(x)
Common derivatives: (eˣ)′ = eˣ, (ln x)′ = 1/x, (sin x)′ = cos x, (cos x)′ = −sin x

Method

Identify the structure: sum, product, quotient or composition.
Apply the power rule to each power of x.
Use the product, quotient or chain rule where needed.
Simplify, and evaluate at the point if asked.

Numerical methods

Bisection: c = (a + b)/2, width after n steps (b − a)/2ⁿ
Newton–Raphson: x_n+1 = xₙ − f(xₙ)/f′(xₙ)
Linear interpolation: y = y₁ + (y₂ − y₁)(x − x₁)/(x₂ − x₁)
Trapezoid rule: T = (h/2)[f₀ + 2f₁ + … + 2f_n−1 + fₙ], h = (b − a)/n
Simpson’s rule (n even): S = (h/3)[f₀ + 4f₁ + 2f₂ + … + 4f_n−1 + fₙ]
Trapezoid error bound: |E| ≤ (b − a)h²M/12, M = max|f″|
Euler: y_n+1 = yₙ + h·f(tₙ, yₙ)
Heun: y_n+1 = yₙ + (h/2)(k₁ + k₂), k₂ = f(tₙ + h, yₙ + hk₁)
RK4: y_n+1 = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄)

Method

For a root: check the sign change for bisection, or compute f and f′ for Newton, and keep six or more digits in a table of iterates.
For a table value: interpolate linearly between the two neighbouring points.
For an integral: find h = (b − a)/n, list f at every node, apply the weights (1, 2, …, 2, 1 for trapezoid; 1, 4, 2, …, 4, 1 for Simpson with even n) and multiply by h/2 or h/3.
For an ODE: step from t₀ with yₙ₊₁ = yₙ + h·f(tₙ, yₙ) (Euler) or the averaged slope (Heun), one row per step, until you reach the requested t.

2Practice problems

1.Differential equationsMedium

Solve y″ + 2y′ + y = 0 with y(0) = 5 and y′(0) = 2. Find y(1).

y″ + 2y′ + y = 0y(0) = 5y′(0) = 2

y(1) =

2.Differential equationsHard

A system obeys y″ + 2y′ + 10y = 10 for t ≥ 0, starting from rest: y(0) = 0, y′(0) = 0. Find the peak value y_max of its step response.

y″ + 2y′ + 10y = 10y(0) = 0y′(0) = 0

y_max =

3.IntegralsMedium

Evaluate I = ∫₀¹ 2x·(x² + 1)³ dx.

f(x) = 2x·(x² + 1)³[0, 1]

I =

4.IntegralsHard

Evaluate I = ∫₀^2.5 x²·eˣ dx.

∫₀^2.5 x²eˣ dx

I =

5.DerivativesMedium

Let f(x) = ln(3x² + 1). Find f′(−2).

f(x) = ln(3x² + 1)x = −2

f′(−2) =

6.DerivativesHard

The tangent line to f(x) = 0.5x² − x − 6 at x = 2 crosses the y-axis at y = b. Find b.

f(x) = 0.5x² − x − 6x = 2

b =

7.Numerical methodsMedium

Approximate ∫₀² eˣ dx with Simpson's rule, n = 2, and compare it with the exact value. Find the absolute error |E| = |S − I|.

f(x) = eˣ[a, b] = [0, 2]n = 2

|E| =

8.Numerical methodsHard

Use Heun's method (improved Euler) with h = 0.2 for y′ = cos t − y, y(0) = 3 to estimate y₃ ≈ y(0.6).

y′ = cos t − yy(0) = 3h = 0.2

y₃ =

3Solutions

1.Differential equations

Answer: y(1) = 4.415

Second-order linear equations: for y″ + b·y′ + c·y = 0 try y = e^rt: this gives the characteristic equation r² + b·r + c = 0. Two real roots r₁ ≠ r₂ give y = A·e^r₁t + B·e^r₂t (overdamped); a double root r gives y = (A + Bt)·e^rt (critically damped); complex roots −α ± jω give y = e^−αt(A cos ωt + B sin ωt) (underdamped, oscillating). The initial values y(0) and y′(0) fix A and B. Written as y″ + 2ζω_ny′ + ω_n²y = 0, ω_n is the natural frequency and ζ the damping ratio: ζ < 1 oscillates. With a constant input, y″ + by′ + cy = c·K, the solution is the steady value K plus the free response.

Double root

(r + 1)² = 0 → r = −1

Constants

A = y(0) = 5

y′(0) = B − 1A → B = 2 + 5 = 7

Evaluate

y(1) = (5 + 7·1)·e⁻¹ = 4.415

Answer

y(1) = 4.415

2.Differential equations

Answer: y_max = 1.351

Steady value and roots

y_ss = 1, r = −1 ± 3j

Constants from the initial values

y(0) = 1 + A = 0 → A = −1

y′(0) = −1A + 3B = 0 → B = −0.3333

Peak

t = π/3 = 1.047

y_max = 1·(1 + e^−1·π/3) = 1.351

Answer

y_max = 1.351

3.Integrals

Answer: I = 3.75

Definite integrals: ∫ₐᵇ f(x) dx is the signed area between the graph and the x-axis from a to b. Find an antiderivative F (with F′ = f) and evaluate F(b) − F(a). Power rule: ∫xⁿ dx = xⁿ⁺¹/(n + 1). The area between two curves is ∫(upper − lower) dx.

Substitution

u = x² + 1, du = 2x dx

I = ∫₁² u³ du

Integrate and evaluate

I = [u⁴/4]₁² = (16 − 1)/4 = 3.75

Answer

I = 3.75

4.Integrals

Answer: I = 37.59

Integration by parts: ∫u·dv = u·v − ∫v·du, from the product rule. Choose u as the factor that gets simpler when you differentiate it (a power of x, or ln x), and dv as the factor you can integrate (eˣ, sin x, cos x, or x when u = ln x). For a definite integral, evaluate u·v at both limits: ∫ₐᵇ u dv = [u·v]ₐᵇ − ∫ₐᵇ v du. A factor x² needs two rounds.

First round

∫x²eˣ dx = x²eˣ − ∫2x eˣ dx

Second round

∫2x eˣ dx = 2x eˣ − 2eˣ

∫x²eˣ dx = eˣ(x² − 2x + 2)

Limits

12.18·(3.25) − 1·2 = 37.59

Answer

I = 37.59

5.Derivatives

Answer: f′(−2) = −0.9231

Differentiation: the derivative f′(x) is the slope of the graph at x. Power rule: (xⁿ)′ = n·xⁿ⁻¹. Product rule: (uv)′ = u′v + uv′. Chain rule: (f(g(x)))′ = f′(g(x))·g′(x). Where f′(x) = 0 the graph has a horizontal tangent: a maximum, minimum or saddle point.

Chain rule for the logarithm

f′(x) = 6x/(3x² + 1)

Substitute

f′(−2) = −12/13 = −0.9231

Answer

f′(−2) = −0.9231

6.Derivatives

Answer: b = −8

Point and slope

f(2) = −6

f′(x) = x − 1 → f′(2) = 1

Tangent line at x = 0

b = f(x₀) − f′(x₀)·x₀ = −6 − 1·2 = −8

Answer

b = −8

7.Numerical methods

Answer: |E| = 0.03167

Numerical integration: split [a, b] into n strips of width h = (b − a)/n with nodes x₀ … xₙ. Trapezoid rule: T = (h/2)[f₀ + 2f₁ + 2f₂ + … + 2f_n−1 + fₙ]. Simpson's rule (n even): S = (h/3)[f₀ + 4f₁ + 2f₂ + 4f₃ + … + 4f_n−1 + fₙ]. The trapezoid error shrinks like h² (bound (b − a)h²·M/12 with M = max|f″|), Simpson's like h⁴, so Simpson is usually far more accurate for the same nodes.

Exact value

I = 6.389056

Simpson's rule

S = (1/3)·(1 + 4·2.718282 + 7.389056) = 6.420728

Error

|E| = |6.420728 − 6.389056| = 0.03167

Answer

|E| = 0.03167

Relative error: 0.4957%.

8.Numerical methods

Answer: y₃ = 2.07

Numerical solution of y′ = f(t, y): Euler's method steps along the slope at the start of each step: y_n+1 = yₙ + h·f(tₙ, yₙ). Heun's method (improved Euler) averages the slope at the start and at an Euler prediction: k₁ = f(tₙ, yₙ), k₂ = f(tₙ + h, yₙ + h·k₁), y_n+1 = yₙ + (h/2)(k₁ + k₂). The classical Runge–Kutta method (RK4) combines four slopes: y_n+1 = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄). The global error of Euler is proportional to h, Heun's to h², RK4's to h⁴.

Heun steps

y₁ = 3 + (0.2/2)(−2 − 1.619933) = 2.638007

y₂ = 2.638007 + (0.2/2)(−1.65794 − 1.385358) = 2.333677

y₃ = 2.333677 + (0.2/2)(−1.412616 − 1.225818) = 2.069833

Graph

y₃ = 2.07

Answer

y₃ = 2.07

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