Differential Equations

Summary of Differential Equations

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1. Foundations

Notation:

• Leibniz: dy/dx
• Lagrange: f’(x), f”(x)
• Newton: ẏ, ÿ (for time derivatives)

Differentials vs Derivatives:

• Derivative dy/dx is defined as a limit (single symbol)
• Differentials dx and dy are separate objects, defined later
• dy = f’(x)·dx (definition)
• Their ratio equals the derivative (by construction)
• This justifies “separating” dy/dx algebraically

Multivariable chain rule: $\frac{dg}{dt}=\frac{\partial g}{\partial x}\frac{dx}{dt}+\frac{\partial g}{\partial y}\frac{dy}{dt}$

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2. First-Order ODEs

Separable:

• Form: dy/dx = f(x)·g(y)
• Method: Separate → ∫(1/g)dy = ∫f dx
• Key: RHS must be product, not sum

Linear:

• Form: y’ + P(x)y = Q(x)
• Method: Integrating factor I = e^(∫P dx)
• Solution: y = (1/I)∫I·Q dx
• No constant in integrating factor; constant appears in final integration

Bernoulli:

• Form: y’ + P(x)y = Q(x)yⁿ
• Substitution: v = y^(1-n)
• Divide by yⁿ first, then substitute
• Transforms to linear equation

Homogeneous (y/x type):

• Form: y’ = F(y/x)
• Substitution: v = y/x, so y = vx, y’ = v’x + v
• Transforms to separable

Exact:

• Form: M(x,y)dx + N(x,y)dy = 0
• Test: ∂M/∂y = ∂N/∂x
• Meaning: ∃F such that dF = Mdx + Ndy
• Solution: F(x,y) = C

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3. Second-Order Linear ODEs

Homogeneous with Constant Coefficients

Form: ay” + by’ + cy = 0

Method: Guess y = e^(rx), get characteristic equation ar² + br + c = 0

Three cases based on discriminant b² - 4ac:

Discriminant	Roots	Solution
> 0	Two real r₁, r₂	y = C₁e^(r₁x) + C₂e^(r₂x)
= 0	Repeated r	y = (C₁ + C₂x)e^(rx)
< 0	Complex α ± βi	y = e^(αx)(C₁cos(βx) + C₂sin(βx))

Why dimension 2:

• Second-order → two initial conditions needed
• Solution space corresponds to R²
• Two independent solutions form a basis

Why xe^(rx) for repeated roots:

• Reduction of order method
• Assume y = v·e^(rx), find v” = 0, so v = Ax + B

Complex roots explained:

• e^(ix) = cos(x) + i·sin(x) (Euler’s formula)
• Real/imaginary parts of complex solution are each solutions (for real coefficients)

Non-Homogeneous

Form: ay” + by’ + cy = g(x)

General solution: y = y_c + y_p

Method 1: Undetermined Coefficients

• Guess y_p based on form of g(x)
• Works for: polynomials, exponentials, sin, cos, products
• If guess matches y_c, multiply by x
• Why multiplying by x works: L[xf] = x·L[f] + (extra terms) = 0 + (extra terms)

Method 2: Variation of Parameters

• Always works
• Assume y_p = u₁y₁ + u₂y₂
• Impose: u₁’y₁ + u₂’y₂ = 0 (simplifies derivatives)
• ODE gives: u₁’y₁’ + u₂’y₂’ = g(x)
• Solve system for u₁’, u₂’, then integrate

Wronskian: $W=y_1{y}_2^{\prime }-y_2{y}_1^{\prime }$

• Measures linear independence
• Used in variation of parameters

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4. Applications

Mixing/Brine Problems

Setup:

• y(t) = amount of substance in tank
• dy/dt = (rate in) - (rate out)

Key formulas:

• Rate in = C₁ · r₁ (concentration × flow rate)
• Rate out = (y/V) · r₂ (current concentration × flow rate)
• V(t) = V₀ + (r₁ - r₂)t

Three cases:

Condition	Volume	Solution shape
r₁ = r₂	constant	exponential decay
r₁ > r₂	grows	1/(a+t)ⁿ
r₁ < r₂	shrinks	(a-t)ⁿ

Newton’s Cooling

$\frac{dT}{dt}=-k(T-T_{ambient})$

Solution: T(t) = T_ambient + (T₀ - T_ambient)e^(-kt)

Circuits

RL circuit (first-order): $L\frac{di}{dt}+Ri=V$

RLC circuit (second-order): $L\frac{d^{2}q}{d{t}^2}+R\frac{dq}{dt}+\frac{q}{C}=V$

Damping cases:

Case	Discriminant	Behavior
Overdamped (sovrasmorzato)	> 0	Smooth decay, no oscillation
Critically damped (smorzamento critico)	= 0	Fastest decay without oscillation
Underdamped (sottosmorzato)	< 0	Oscillates while decaying

Springs

Hooke’s Law: F = -kx (spring force)

Newton’s Law: F = ma = mx”

Combined: mx” + kx = 0

With damping: mx” + cx’ + kx = 0

Same three cases as circuits.

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5. Qualitative Analysis

Direction Fields

• At each point (t, y), draw slope = f(t, y)
• Direction vector: (1, f(t,y))
• Solution curves are tangent to the field

Autonomous Equations

Form: dy/dt = f(y) (no explicit t dependence)

Key property: Same y → same slope, regardless of t

Direction field: Horizontal bands of constant slope

Equilibrium Solutions

Definition: Values where f(y) = 0 (constant solutions)

Classification:

• Stable (attractor): Both sides push toward equilibrium
• Unstable (repeller): Both sides push away
• Semi-stable: One side toward, one side away

Method: Check sign of f(y) just above and below equilibrium

Euler’s Method

$y_n=y_{n-1}+f(t_{n-1},y_{n-1})\cdot \Delta t$

• Numerical approximation
• Follow direction field in small steps
• n = step number, not time value

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6. Power Series Solutions

When to Use

• At ordinary points (punto ordinario): P(x) and Q(x) analytic
• At singular points (punto singolare): Need Frobenius method

Method

Step 1: Write y = Σcₙxⁿ and derivatives as series

Step 2: Substitute into ODE

Step 3: Get series “in phase” (same starting power of x)

• May need to pull terms outside sum

Step 4: Re-index to match powers

Step 5: Combine and set coefficient of each xᵏ to zero

Step 6: Get recurrence relation, solve for cₙ

Key Techniques

Re-indexing example:

• If sum has x^(n-1), let k = n-1, n = k+1
• Subscript cₙ becomes c_{k+1}

Series multiplication:

• For cos(x)·y, multiply term by term
• Collect like powers of x

Common Results

$e^x=\sum \frac{x^n}{n!}$

$\sin (x)=\sum \frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$

$\cos (x)=\sum \frac{(-1{)}^n{x}^{2n}}{(2n)!}$

$\frac{1}{1-u}=\sum u^n(|u|<1)$

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7. Solution Process Summary

First-order:

1. Is it separable? → Separate and integrate
2. Is it linear? → Integrating factor
3. Is it Bernoulli? → Substitute v = y^(1-n)
4. Is it homogeneous (y/x)? → Substitute v = y/x
5. Is it exact? → Find F where dF = 0

Second-order linear:

1. Solve homogeneous first (characteristic equation)
2. If non-homogeneous, find y_p via:
   • Undetermined coefficients (if g(x) is nice)
   • Variation of parameters (always works)
3. General solution: y = y_c + y_p
4. Apply initial conditions last