7.2 Verifying Solutions for Differential Equations

• 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 \( y = e^{3x} \)?

  $3e^{3x}$
• 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 \( y = e^{3x} \)?

  $3e^{3x}$
• 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