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″ + 8y′ + 15y = 0 with y(0) = 4 and y′(0) = -2. Find y(1.5).

y″ + 8y′ + 15y = 0y(0) = 4y′(0) = -2

y(1.5) =

2.Differential equationsHard

Solve the initial value problem y′ + 3y/x = x² for x > 0, with y(1) = −1. Find y(2).

y′ + 3y/x = x²y(1) = −1

y(2) =

3.IntegralsMedium

Evaluate I = ∫₀^π x·cos x dx (x in radians).

∫₀^π x·cos x dx

I =

4.IntegralsHard

Evaluate I = ∫₀¹ x²·eˣ dx.

∫₀¹ x²eˣ dx

I =

5.DerivativesMedium

Let f(x) = x·sin(2x), with x in radians. Find f′(1.5).

f(x) = x·sin(2x)x = 1.5 rad

f′(1.5) =

6.DerivativesHard

Let f(x) = x³ + 1.5x². Find the x-coordinate x of the local maximum.

f(x) = x³ + 1.5x²

x =

7.Numerical methodsMedium

Approximate ∫₀^1.5 sin(x²) dx with the trapezoid rule using n = 5 strips. Give T.

f(x) = sin(x²)[a, b] = [0, 1.5]n = 5

T =

8.Numerical methodsHard

Use Heun's method (improved Euler) with h = 0.2 for y′ = e^−ty, y(0) = 0.5 to estimate y₃ ≈ y(0.6).

y′ = e^−tyy(0) = 0.5h = 0.2

y₃ =

3Solutions

1.Differential equations

Answer: y(1.5) = 0.09722

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.

Characteristic roots

(r + 3)(r + 5) = 0 → r = −3, −5

Constants from the initial values

A + B = 4, −3A − 5B = -2

A = 9, B = −5

Evaluate

y(1.5) = 9·e^−4.5 + (−5)·e^−7.5 = 0.09722

Answer

y(1.5) = 0.09722

2.Differential equations

Answer: y(2) = 1.188

First-order differential equations: dy/dt = k·y has the solution y = y₀·e^kt (growth for k > 0, decay for k < 0); a half-life T gives k = ln 2/T. Newton's law of cooling, dT/dt = −k·(T − Tₐ), gives T(t) = Tₐ + (T₀ − Tₐ)·e^−kt. A linear equation y′ + p(x)·y = q(x) is solved with the integrating factor μ = e^{∫p dx}: (μy)′ = μq, so μy = ∫μq dx + C.

Integrating factor

μ = e^{∫3/x dx} = x³

Integrate

(x³y)′ = x⁵ → x³y = 1x⁶/6 + C

Initial condition

−1 = 1/6 + C → C = −1.167

Evaluate

y(2) = (1·2⁶/6 + (−1.167))/2³ = 1.188

Answer

y(2) = 1.188

3.Integrals

Answer: I = −2

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.

Choose u and dv

u = x, du = dx; dv = cos x dx, v = sin x

Antiderivative

∫x cos x dx = x sin x − ∫sin x dx = x sin x + cos x

Limits

(π·sin π + cos π) − (0 + 1) = −2

Answer

I = −2

4.Integrals

Answer: I = 0.7183

First round

∫x²eˣ dx = x²eˣ − ∫2x eˣ dx

Second round

∫2x eˣ dx = 2x eˣ − 2eˣ

∫x²eˣ dx = eˣ(x² − 2x + 2)

Limits

2.718·(1) − 1·2 = 0.7183

Answer

I = 0.7183

5.Derivatives

Answer: f′(1.5) = −2.829

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.

Product and chain rule

f′(x) = sin(2x) + 2x·cos(2x)

Substitute

f′(1.5) = 0.1411 + 3·(−0.99) = −2.829

Answer

f′(1.5) = −2.829

6.Derivatives

Answer: x = −1

Derivative

f′(x) = 3x² + 3x = 3·(x² + x)

Solve f′(x) = 0

x = −1 or x = 0

Which one is which

f″(x) = 6x + 3: negative at a maximum, positive at a minimum.

f″(−1) = −3, f″(0) = 3

x = −1

Answer

x = −1

7.Numerical methods

Answer: T = 0.7641

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.

Nodes and values

h = 0.3

Σ wᵢfᵢ = 0 + 2·0.0898785 + 2·0.352274 + 2·0.724287 + 2·0.991458 + 0.778073 = 5.09387

Trapezoid rule

T = (0.3/2)·5.09387 = 0.7641

Answer

T = 0.7641

This integral has no elementary antiderivative, which is exactly when a numerical rule is needed.

8.Numerical methods

Answer: y₃ = 0.9716

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 = 0.5 + (0.2/2)(1 + 0.869358) = 0.686936

y₂ = 0.686936 + (0.2/2)(0.871633 + 0.708571) = 0.844956

y₃ = 0.844956 + (0.2/2)(0.713208 + 0.552911) = 0.971568

Graph

y₃ = 0.9716

Answer

y₃ = 0.9716

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