7.2 Verifying Solutions for Differential Equations

Cards (70)

  • Match the type of differential equation with its characteristic:
    ODE ↔️ Single independent variable
    PDE ↔️ Multiple independent variables
    Total derivatives ↔️ Derivatives in ODEs
    Partial derivatives ↔️ Derivatives in PDEs
  • The function \( y = e^{3x} \) is a solution to the differential equation \( \frac{dy}{dx} = 3y \).

    True
  • Ordinary Differential Equations (ODEs) involve multiple independent variables.
    False
  • Steps to verify a solution to a differential equation:
    1️⃣ Find the necessary derivatives
    2️⃣ Substitute into the equation
    3️⃣ Simplify and check
  • Match the component of a differential equation with its description:
    Equation Type ↔️ ODE or PDE
    Unknown Function ↔️ \( y(x) \) or \( u(x, t) \)
    Derivatives ↔️ One or more derivatives of the unknown function
  • A solution to a differential equation must satisfy the equation for all values in the given interval.

    True
  • Steps for verifying a solution to a differential equation:
    1️⃣ Find the necessary derivatives
    2️⃣ Substitute into the equation
    3️⃣ Simplify and check
  • Verifying a solution involves checking if both sides of the equation are equal
  • The derivative of \( e^{3x} \) with respect to \( x \) is \( 3e^{3x} \).

    True
  • Steps to verify a function as a solution to a differential equation:
    1️⃣ Find the necessary derivatives
    2️⃣ Substitute into the equation
    3️⃣ Simplify and check
  • What is the first step in verifying that a function is a solution to a differential equation?
    Find the necessary derivatives
  • What is the derivative of \( y = e^{3x} \) with respect to \( x \)?
    \( 3e^{3x} \)
  • Substituting derivatives into a differential equation is a key step in verifying its solution.

    True
  • A differential equation relates a function to its rate of change with respect to one or more independent
  • A solution to a differential equation must satisfy the equation for all values of the independent variable within a given interval.

    True
  • What type of equation is \( \frac{dy}{dx} = 3y \)?
    Ordinary Differential Equation
  • What is the derivative of \( y = e^{3x} \)?
    3e3x3e^{3x}
  • To verify a solution to a differential equation, you must simplify both sides and check for equality
  • If the left-hand side equals the right-hand side after simplification, the proposed solution is valid.

    True
  • What is the derivative of \( y = e^{3x} \)?
    3e3x3e^{3x}
  • What is the first step in verifying a solution to a differential equation?
    Find necessary derivatives
  • When verifying a solution, you must ensure that both sides of the equation are equal
  • If the left-hand side and right-hand side are equal after simplification, the proposed solution is valid.

    True
  • Match the aspect with its description:
    Form of LHS ↔️ Derivatives of the solution
    Form of RHS ↔️ Original solution or constants
    Goal of LHS ↔️ Match the RHS
  • What is the conclusion if the left-hand side does not equal the right-hand side?
    Not a valid solution
  • A differential equation relates a function to its rate of change with respect to one or more independent variables
  • A solution to a differential equation must satisfy the equation for all values of the independent variable within a given interval
  • Ordinary Differential Equations (ODEs) involve a single independent variable
  • If both sides of the simplified equation are equal, the solution is verified
  • A solution to a differential equation must satisfy the equation within a given interval
  • What is the purpose of verifying a solution to a differential equation?
    To ensure it satisfies the equation
  • What is the differential equation that \( y = e^{3x} \) is a solution to?
    \( \frac{dy}{dx} = 3y \)
  • What is the first step in verifying that a function is a solution to a differential equation?
    Find the necessary derivatives
  • What is the final step in verifying that a function is a solution to a differential equation?
    Simplify and check
  • The function \( y = e^{3x} \) is a solution to the differential equation \( \frac{dy}{dx} = 3y \)
    True
  • What condition must be satisfied for a function to be a verified solution to a differential equation?
    The equation must hold
  • What is a differential equation?
    An equation involving derivatives
  • Partial Differential Equations (PDEs) involve multiple independent variables
  • Match the component of a differential equation with its description:
    Equation Type ↔️ ODE or PDE
    Unknown Function ↔️ \( y(x) \) or \( u(x, t) \)
    Derivatives ↔️ One or more derivatives
    Objective ↔️ To solve for \( y(x) \) or \( u(x, t) \)
  • After substituting the solution and its derivatives, you must simplify and check if both sides are equal