Differential Equations
What is Differential Equations?
An equation that relates an unknown function to its derivatives.
Key formula / rule: Integrating Factor for First-Order Linear ODE
Key points
- Understand the definition and order of differential equations.
- Identify different types of differential equations (e.g., linear, non-linear, separable).
- Solve first-order linear and separable ordinary differential equations (ODEs).
- Solve second-order linear ODEs with constant coefficients.
Common exam trap
Incorrectly identifying the order or linearity of an equation.
Definitions
- Term
Differential Equation
- Meaning
An equation that relates an unknown function to its derivatives.
- Term
Ordinary Differential Equation (ODE)
- Meaning
A differential equation containing only ordinary derivatives of an unknown function of a single independent variable.
- Term
Order of a Differential Equation
- Meaning
The order of the highest derivative present in the equation.
- Term
Linear Differential Equation
- Meaning
A differential equation in which the dependent variable and its derivatives appear linearly.
- Term
Integrating Factor
- Meaning
A function that, when multiplied by a non-exact differential equation, makes it exact or simplifies it for solution.
Learning objectives
Understand the definition and order of differential equations.
Identify different types of differential equations (e.g., linear, non-linear, separable).
Solve first-order linear and separable ordinary differential equations (ODEs).
Solve second-order linear ODEs with constant coefficients.
Apply differential equations to model simple physical phenomena.
Formulae
- Name
Integrating Factor for First-Order Linear ODE
- Note
Used for solving equations of the form dy/dx + P(x)y = Q(x)
- Expression
IF = e\int P(x) dx
- Name
General Solution for First-Order Linear ODE
- Note
Where IF is the integrating factor.
- Expression
y = \frac{1}{IF} \left( \int Q(x) \· IF dx + C \right)
- Name
Characteristic Equation for Second-Order Linear ODE with Constant Coefficients
- Note
For the equation ay'' + by' + cy = 0
- Expression
ar2 + br + c = 0
Prerequisites
Calculus (Differentiation and Integration)
Algebra (Solving equations)
Common mistakes
Incorrectly identifying the order or linearity of an equation.
Errors in applying the integrating factor for first-order linear ODEs.
Mistakes in solving the characteristic equation for second-order ODEs.
Forgetting to include the constant of integration when solving.
Confusing initial conditions with boundary conditions.
Keywords
Differential Equation
ODE
First Order
Second Order
Linear ODE
Separable ODE
Integrating Factor
Characteristic Equation
Homogeneous ODE
Constant Coefficients
Practice preview
Which of the following differential equations is non-linear?…
easy
The general solution of the differential equation (x^2 + y^2)dx - 2xydy = 0 is:…
medium
Which of the following conditions must be satisfied for the differential equation M(x,y)dx + N(x,y)dy = 0 to be exact?…
medium