7.2 Verifying Solutions for Differential Equations

• Variables in a differential equation can be independent or dependent
• Match the type of differential equation with its description:

  ODEs ↔️ One independent variable

  PDEs ↔️ Multiple independent variables
• A differential equation relates a function to its derivatives
• Checking initial conditions involves substituting initial values into the proposed solution.


  True
• What is a differential equation?

  Equation with derivatives
• A Partial Differential Equation (PDE) involves multiple independent variables
• Steps to classify differential equations by type:

  1️⃣ Identify the independent variables

  2️⃣ Determine the order of the derivatives

  3️⃣ Check if the equation is linear or non-linear

  4️⃣ Classify as ODE or PDE
• Direct substitution checks if the proposed solution satisfies the differential equation.


  True
• What is a differential equation?

  Relates a function to derivatives
• A solution to a differential equation is a function that, when substituted, satisfies the equation
• PDEs involve partial derivatives.


  True
• Differential equations involve both independent and dependent variables
• Match the type of differential equation with its characteristic:

  ODEs ↔️ Ordinary derivatives

  PDEs ↔️ Partial derivatives
• ODEs involve only one independent variable.


  True
• What is the first step in verifying a solution to a differential equation?

  Differentiate the solution
• The function \( y = x^2 + x + C \) is a solution to \( y' = 2x + 1 \)


  True
• Match the example with the type of differential equation:

  ODEs ↔️ y' = 2x + 1

  PDEs ↔️ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}
• What is the derivative of \( y = x^2 + 3 \)?

  y' = 2x
• Derivatives in differential equations can be first-order or higher-order.


  True
• What is a solution to a differential equation?

  Function satisfying the equation
• What type of derivatives are used in PDEs?

  Partial derivatives
• Steps to verify a solution for a differential equation:

  1️⃣ Differentiate the proposed solution

  2️⃣ Substitute the solution and derivatives into the equation

  3️⃣ Simplify both sides to check for equality
• Ordinary Differential Equations (ODEs) involve functions of one variable.

  True
• What are the key components of a differential equation?

  Variables, derivatives, relationships
• Steps to verify a solution using direct substitution:

  1️⃣ Substitute the proposed solution and its derivatives into the differential equation

  2️⃣ Verify that both sides of the equation are equal
• The function \( y = x^2 + 3 \) is a valid solution to \( y' = 2x \) with initial condition \( y(0) = 3
• What does an Ordinary Differential Equation (ODE) involve?

  One independent variable
• To verify a solution to a differential equation, you substitute the function and its derivatives
• When verifying a solution using initial conditions, you substitute the initial values into the proposed solution