2.8 Differential Equations

    Cards (193)

    • There are two main types of differential equations: ODEs and PDEs
      True
    • Ordinary Differential Equations (ODEs) involve functions of a single variable
    • The example equation `dy/dx + 2y = x` is an Ordinary Differential Equation (ODE).

      True
    • What is the key difference between Ordinary Differential Equations and Partial Differential Equations?
      Number of independent variables
    • PDEs are used to model complex systems such as heat diffusion.
      True
    • What type of derivatives do PDEs involve?
      Partial derivatives
    • The separation of variables method is used to solve first-order differential equations.
    • Match the type of differential equation with its feature:
      ODE ↔️ Single independent variable
      PDE ↔️ Multiple independent variables
    • What does the order of a differential equation refer to?
      Highest derivative present
    • In the equation `d²y/dx² + 2y = x`, the degree is 2
    • Solve the differential equation `dy/dx = 2xy` using the separation of variables method.
      y = Ae^(x²)
    • A homogeneous differential equation is one where the dependent variable and its derivative appear only as a product with the independent variable `x`.

      True
    • An integrating factor is used to solve a first-order linear differential equation.
    • What is the general form of a second-order differential equation with constant coefficients?
      a(d²y/dx²) + b(dy/dx) + cy = 0</latex>
    • Match the root types with their corresponding general solutions for a second-order homogeneous ODE:
      Distinct real roots ↔️ y=y =A A *e(m1x)+ e^(m1x) +B B *e(m2x) e^(m2x)
      Repeated real roots ↔️ y=y =(A+Bx)e(mx) (A + Bx)e^(mx)
      Complex conjugate roots ↔️ y=y =e(αx)(Acos(βx)+ e^(αx)(A * cos(βx) +B B *sin(βx)) sin(βx))
    • What determines the general solution of a homogeneous second-order ODE?
      Nature of the roots
    • The general solution for complex conjugate roots `α ± βi` is `y = e^(αx)(A*cos(βx) + B*sin(βx))`, where `α` represents the real part and `β` represents the imaginary part.
    • Ordinary differential equations involve only ordinary derivatives.

      True
    • Match the type of differential equation with its application:
      Ordinary Differential Equations (ODEs) ↔️ Population growth
      Partial Differential Equations (PDEs) ↔️ Heat diffusion
    • What is the order of the differential equation ²y/dx² + dy/dx + y = x?
      2
    • The degree of a differential equation is the highest power of the highest derivative.
      True
    • Steps to solve a first-order differential equation using separation of variables:
      1️⃣ Separate the variables
      2️⃣ Integrate both sides
      3️⃣ Solve for the dependent variable
    • A differential equation is used to model real-world phenomena such as population growth, radioactive decay, and fluid dynamics
    • What is an example of a Partial Differential Equation (PDE)?
      \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2}</latex>
    • What is an example of a real-world system modeled by a Partial Differential Equation (PDE)?
      Heat diffusion
    • What does an ordinary differential equation (ODE) involve?
      Single variable and ordinary derivatives
    • What is the purpose of a differential equation?
      Model real-world phenomena
    • The degree of a differential equation is the highest power of its dependent variable and derivatives.

      True
    • What is the first step in solving a first-order differential equation using the separation of variables method?
      Separate the variables
    • What does the order of a differential equation refer to?
      Highest derivative present
    • In the equation `dy/dx + 2y = x`, the order is 1
    • The separation of variables method involves arranging terms with the dependent variable and its derivative on one side and terms with the independent variable on the other.

      True
    • The separation of variables method is not applicable if the variables cannot be separated
    • Steps to solve a homogeneous differential equation
      1️⃣ Identify the equation as homogeneous
      2️⃣ Substitute a new variable
      3️⃣ Separate the variables
      4️⃣ Integrate both sides
      5️⃣ Solve for the original variable
    • The integrating factor `μ(x)` is defined as e(P(x)dx)e^(∫P(x) dx) where `P(x)` is from the equation `dy/dx + P(x)y = Q(x)`.

      True
    • For a second-order homogeneous ODE, the characteristic equation is obtained by substituting y = e^(mx) into the differential equation.
    • What is the general form of a second-order differential equation?
      a(d2y/dx2)+a(d²y / dx²) +b(dy/dx)+ b(dy / dx) +cy= cy =0 0
    • The general solution for distinct real roots `m1` and `m2` is `y = A*e^(m1x) + B*e^(m2x)`.

      True
    • Second-order ODEs are used in modeling physical phenomena like simple harmonic motion.

      True
    • What type of derivatives do partial differential equations involve?
      Partial derivatives