Ordinary Differential Equation (ODE)
What is Ordinary Differential Equation (ODE)?
An equation involving an unknown function of a single independent variable and its derivatives.
Key formula / rule: Linear First-Order ODE (General Form)
Key points
- Define and classify ODEs based on order, °, linearity, and homogeneity.
- Solve various types of first-order ODEs using appropriate methods.
- Solve higher-order linear ODEs with constant coefficients by finding CF and PI.
- Apply initial and boundary conditions to determine particular solutions.
Common exam trap
Incorrectly identifying the order or ° of an ODE.
Definitions
- Term
Ordinary Differential Equation (ODE)
- Meaning
An equation involving an unknown function of a single independent variable and its derivatives.
- Term
Order of an ODE
- Meaning
The order of the highest derivative present in the differential equation.
- Term
Degree of an ODE
- Meaning
The power of the highest order derivative when the differential equation is expressed as a polynomial in derivatives.
- Term
Linear ODE
- Meaning
An ODE where the dependent variable and its derivatives appear only in the first ° and are not multiplied together.
- Term
Homogeneous ODE (for linear)
- Meaning
A linear ODE where every term contains the dependent variable or one of its derivatives (i.e., the right-hand side is zero).
- Term
General Solution
- Meaning
A solution to an ODE that contains arbitrary constants equal to the order of the equation.
- Term
Particular Solution
- Meaning
A solution obtained from the general solution by assigning specific values to the arbitrary constants using initial or boundary conditions.
- Term
Integrating Factor
- Meaning
A function by which a non-exact differential equation can be multiplied to make it exact, or to solve a linear first-order ODE.
- Term
Complementary Function (CF)
- Meaning
The general solution of the homogeneous part of a non-homogeneous linear ODE.
- Term
Particular Integral (PI)
- Meaning
Any specific solution of the non-homogeneous part of a non-homogeneous linear ODE.
Learning objectives
Define and classify ODEs based on order, °, linearity, and homogeneity.
Solve various types of first-order ODEs using appropriate methods.
Solve higher-order linear ODEs with constant coefficients by finding CF and PI.
Apply initial and boundary conditions to determine particular solutions.
Understand the physical interpretation and real-world applications of ODEs in engineering contexts.
Formulae
- Name
Linear First-Order ODE (General Form)
- Note
P(x) and Q(x) are functions of x.
- Expression
dy/dx + P(x)y = Q(x)
- Name
Integrating Factor (Linear First-Order ODE)
- Note
Used to solve linear first-order ODEs.
- Expression
IF = e^∫P(x)dx
- Name
Solution of Linear First-Order ODE
- Note
C is the constant of integration.
- Expression
y * IF = ∫(Q(x) * IF)dx + C
- Name
Exact ODE Condition
- Note
Condition for an ODE to be exact.
- Expression
M(x,y)dx + N(x,y)dy = 0 if ∂M/∂y = ∂N/∂x
- Name
Solution of Exact ODE
- Note
Integrate M with respect to x (y constant), then integrate terms of N not containing x with respect to y.
- Expression
∫M(x,y)dx (y const) + ∫(terms in N not containing x)dy = C
- Name
Homogeneous ODE Substitution
- Note
Used for ODEs of the form dy/dx = f(y/x).
- Expression
y = vx => dy/dx = v + x(dv/dx)
- Name
Bernoulli's Equation Transformation
- Note
Transforms Bernoulli's equation into a linear first-order ODE.
- Expression
z = y^(1-n) for dy/dx + P(x)y = Q(x)yn
- Name
CF for Real Distinct Roots (m1, m2)
- Note
For auxiliary equation roots m1 ≠ m2.
- Expression
c1e^(m1x) + c2e^(m2x)
- Name
CF for Real Repeated Roots (m, m)
- Note
For auxiliary equation roots m1 = m2 = m.
- Expression
(c1 + c2x)e^(mx)
- Name
CF for Complex Conjugate Roots (α ± iβ)
- Note
For auxiliary equation roots α ± iβ.
- Expression
e^(αx)(c1cos(βx) + c2sin(βx))
- Name
PI for R(x) = e^(ax)
- Note
Method of undetermined coefficients for exponential RHS. D is the differential operator d/dx.
- Expression
e^(ax) / f(D) where D=a (if f(a) ≠ 0); if f(a)=0, x * e^(ax) / f'(D) where D=a
- Name
PI for R(x) = sin(ax) or cos(ax)
- Note
Method of undetermined coefficients for trigonometric RHS. D is the differential operator d/dx.
- Expression
sin(ax) / f(D2) where D2 = -a2 (if f(-a2) ≠ 0); if f(-a2)=0, x * sin(ax) / f'(D2) where D2 = -a2
Prerequisites
Basic differentiation and integration techniques from calculus.
Algebraic skills, including solving polynomial equations and partial fractions.
Understanding of complex numbers for roots of auxiliary equations.
Basic knowledge of functions and their properties.
Common mistakes
Incorrectly identifying the order or ° of an ODE.
Errors in integration or algebraic manipulation during solution steps.
Misapplying conditions for exactness or integrating factors.
Incorrectly finding roots of the auxiliary equation for higher-order ODEs.
Errors in determining the form of the Particular Integral (PI), especially in cases of resonance (when the RHS term is part of the CF).
Keywords
Differential Equation
ODE
Order
Degree
Linear
Non-linear
Homogeneous
Exact
Integrating Factor
Variable Separable
Bernoulli
Complementary Function
Particular Integral
Auxiliary Equation
Initial Value Problem
Boundary Value Problem
Practice preview
The particular integral of the differential equation d^2y/dx^2 - 3(dy/dx) + 2y = e^(3x) is:…
hard
Which of the following is a linear ordinary differential equation?…
easy
The solution of the differential equation (y^2 - 2xy)dx + (2xy - x^2)dy = 0 is:…
medium