Matrix Algebra and Differential Equations Essentials

Posted on Sep 2, 2025 in Mathematics

Matrix Rank

  • Echelon Form: rank = number of non-zero rows
  • Normal Form: reduce to identity matrix (I); rank = number of 1s on diagonal

Matrix Forms

  • Echelon Form: zeros below pivots
  • Normal Form (RREF): pivots = 1 and only non-zero in column

Systems of Equations

  • Form augmented matrix [A|B], row reduce
  • Consistent if no row like [0 0 0 | b≠0]
  • Unique: rank A = rank [A|B] = number of variables
  • Infinite: rank A = rank [A|B] < number of variables
  • No solution: rank A ≠ rank [A|B]

Linear Dependence/Independence

  • Vectors u, v, w are dependent if: a·u + b·v + c·w = 0 has a non-zero solution
  • Solve the system of equations for a, b, c

Characteristic Polynomial

  • For matrix A: find |A – λI| = 0
  • Gives a polynomial in λ
  • Roots = eigenvalues

Eigenvalues

  • Solve the characteristic polynomial: |A – λI| = 0
  • λ = eigenvalues

Cayley-Hamilton Theorem

  • If |A – λI| = λ² + aλ + b, then: A² + aA + bI = 0

Partial Differentiation Basics

First-order partial derivatives:

∂z/∂x, ∂z/∂y

Second-order mixed partial derivatives (Clairaut’s Theorem, if f is continuous):

∂²z/∂x∂y = ∂²z/∂y∂x

Euler’s Theorem (Homogeneous Functions)

If u = f(x, y) is homogeneous of degree n, then:

x ∂u/∂x + y ∂u/∂y = n u

Euler’s Deduction Formulas

For u = f(x, y) homogeneous of degree n:

First Deduction:

x ∂u/∂x + y ∂u/∂y = n u

Second Deduction:

x² ∂²u/∂x² + 2xy ∂²u/∂x∂y + y² ∂²u/∂y² = n(n-1)u

Total Derivative

If u = f(x, y), and x = x(t), y = y(t), then:

du/dt = ∂u/∂x ⋅ dx/dt + ∂u/∂y ⋅ dy/dt

Errors

Absolute Error:

Δx = x(measured) – x(true)

Relative Error:

Δx / x(true)

Percentage Error:

(Δx / x(true)) × 100%

Error in Volume (Example: Cylinder)

For volume V = πr²h:

ΔV/V = 2(Δr/r) + (Δh/h)

Multiply by 100 to get percentage error.

Order & Degree of Differential Equations

  • Order: The highest derivative in the equation.
  • Degree: The power of the highest derivative after removing roots/fractions.

Formation of a Differential Equation

Given: A family of curves with constants.

Steps:

  1. Differentiate as needed.
  2. Eliminate constants.

Example 1:

y = mx
⇒ dy/dx = m
⇒ y = x(dy/dx)

Example 2:

x² + y² = r²
⇒ 2x + 2y(dy/dx) = 0
⇒ x + y(dy/dx) = 0

Variable Separable Method

  • Rearrange into the form: dy/dx = f(x) g(y)
  • Separate variables: (1/g(y)) dy = f(x) dx
  • Integrate both sides.

Common Integrals:

∫ dx/(1 + x²) = tan⁻¹ x
∫ eˣ dx = eˣ
∫ sin x dx = -cos x
∫ cos x dx = sin x
∫ tan x dx = ln | sec x |

Linear Differential Equation

  • Standard form: dy/dx + P(x)y = Q(x)
  • Steps:
    1. Find Integrating Factor (IF): IF = e^(∫ P(x) dx)
    2. Multiply all terms by IF.
    3. Left-hand side becomes: d/dx (IF ⋅ y)
    4. Integrate both sides.

Exact Differential Equations

  • Standard form: M(x, y) dx + N(x, y) dy = 0
  • Check Exactness: (∂M/∂y) = (∂N/∂x)
  • If exact:
    1. Integrate M with respect to x.
    2. Add an unknown function of y.
    3. Differentiate the result with respect to y and match with N.
    4. Solve for the unknown function of y, then combine.
    5. Final solution: F(x, y) = C

Second-order Linear DE

( d²y/dx² + a(dy/dx) + by = f(x) )

  • If ( f(x) = 0 ): Homogeneous → Find Complementary Solution (C.F.)
  • If ( f(x) ≠ 0 ): Non-Homogeneous → Find Particular Solution (P.I.)

Step 1: Complementary Solution (C.F.)

Form the Auxiliary Equation (A.E.): ( r² + ar + b = 0 )

Solve based on the discriminant:

1️⃣ Real & Distinct Roots ( r₁, r₂ ):

( y_c = C₁ e^{r₁ x} + C₂ e^{r₂ x} )

2️⃣ Real & Repeated Root ( r ):

( y_c = (C₁ + C₂ x)e^{rx} )

3️⃣ Complex Roots ( r = α ± iβ ):

( y_c = e^{α x} (C₁ cos β x + C₂ sin β x) )

Step 2: Particular Solution (P.I.)

Trial forms based on RHS ( f(x) ):

RHS ( f(x) )Trial for ( yₚ ) (if not in C.F.)
( e^{ax} )( A e^{ax} )
( xⁿ )( A xⁿ + B xⁿ⁻¹ + … )
( sin(ax), cos(ax) )( A cos(ax) + B sin(ax) )
( e^{ax} cos(bx), e^{ax} sin(bx) )( e^{ax} (A cos bx + B sin bx) )

🚨 If trial solution is in C.F., multiply by ( x ), or ( x² ), etc.

Derivatives Needed for ( yₚ )

For ( yₚ = A e^{ax} ):

  • ( yₚ’ = a A e^{ax} )
  • ( yₚ” = a² A e^{ax} )

For ( yₚ = A cos ax + B sin ax ):

  • ( yₚ’ = -a A sin ax + a B cos ax )
  • ( yₚ” = -a² A cos ax – a² B sin ax )

Final Solution

( y = y_c + y_p )

Where:

  • ( y_c ) is the complementary solution
  • ( y_p ) is the particular solution

Statistics & Probability Essentials

Mean ( x̄ )

Mean:
x̄ = Σ(f × x) / Σf

Standard Deviation ( σ )

Standard Deviation:
σ = √(Σ f (x – x̄)² / Σ f)

Mean Deviation (about Mean)

Mean Deviation = Σ f | x – x̄ | / Σ f

Median (Ungrouped Data)

Steps:

  1. Arrange data in order.
  2. If odd number of values: Median = Middle value.
  3. If even number of values: Median = ((n/2)th value + (n/2+1)th value) / 2

Mode (Ungrouped Data)

  • The value that occurs most frequently in the dataset.

Probability

Probability = Favorable Outcomes / Total Outcomes