7.2 Verifying Solutions for Differential Equations

• What is a differential equation?

  An equation relating derivatives
• An ordinary differential equation (ODE) involves a single independent variable
• Partial differential equations (PDEs) involve multiple independent variables.

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

  Ordinary Differential Equation ↔️ dy/dx = 3x + 2

  Partial Differential Equation ↔️ ∂u/∂x + ∂u/∂y = x^2y
• What are some physical phenomena modeled by differential equations?

  Population growth, chemical reactions
• Steps to verify a solution for a differential equation:

  1️⃣ Find the necessary derivatives

  2️⃣ Substitute the function and its derivatives into the equation

  3️⃣ Simplify the equation
• If substitution results in an always true equation, the function is a solution.


  True
• The derivative of \( y = \frac{3}{2}x^2 + 2x + C \) is \( y' = 3x + 2 \), which satisfies the differential equation \( y' = 3x + 2 \). This process is called substitution
• What is an example of an ordinary differential equation?

  dy/dx = 3x + 2
• What is an example of a partial differential equation?

  ∂u/∂x + ∂u/∂y = x^2y
• Steps to verify a solution for a differential equation:

  1️⃣ Find the necessary derivatives

  2️⃣ Substitute the function and its derivatives

  3️⃣ Simplify the equation
• How do you verify that \( y = \frac{3}{2}x^2 + 2x + C \) is a solution to \( y' = 3x + 2 \)?

  Find y', substitute, simplify
• If the substitution results in a false equation, the proposed solution is incorrect.


  True
• To verify a solution for a first-order differential equation, you must first find its derivative
• Why is \( y = \frac{x^2}{2} + x + C \) a solution to \( y' = x + 1 \)?

  It satisfies the equation
• Substitution is the primary technique for verifying solutions to differential equations.

  True
• What are the steps for verifying a solution using the substitution method?

  Find derivatives, substitute, simplify
• The function \( y = x^2 + C \) is a solution to the differential equation \( y' = 2x \) because its derivative is also \( 2x \), demonstrating the process of verification
• Verifying solutions is crucial in understanding differential equations.

  True
• What is a differential equation?

  Relates function with its derivatives
• There are two main types of differential equations: ordinary differential equations and partial
• An ordinary differential equation (ODE) contains derivatives with respect to multiple independent variables.

  False
• What is an example of a partial differential equation (PDE)?

  $\frac{\partial u}{\partial x} +$$\frac{\partial u}{\partial y} =$$x^{2}y$
• Match the differential equation with its proposed solution:

  y' = 3x + 2 ↔️ y = \frac{3}{2}x^2 + 2x + C

  y' = x + 1 ↔️ y = \frac{x^2}{2} + x + C
• Verifying a solution for a differential equation requires showing that the given function and its derivatives satisfy the equation.

  True
• Steps to verify a solution for a first-order differential equation:

  1️⃣ Find the derivative of the proposed solution

  2️⃣ Substitute the solution and its derivative into the differential equation

  3️⃣ Simplify the equation to check if it holds true
• What does it mean if the simplified equation is always true when verifying a solution?

  The solution is correct
• What is the first step in verifying a proposed solution to a differential equation?

  Find the necessary derivatives
• If the simplified equation is always true, then the proposed solution is correct
• The equation 3x + 2 = 3x + 2 holds true, indicating the proposed solution is correct.

  True
• To verify a solution to a differential equation, you must first find the necessary derivatives
• In an incorrect solution example, what happens when the solution is substituted into the differential equation?

  The equation does not hold true
• For a second-order differential equation, you need to find both the first and second derivatives
• Steps to verify a solution for a second-order differential equation

  1️⃣ Find the first and second derivatives of the proposed solution

  2️⃣ Substitute the solution and its derivatives into the differential equation

  3️⃣ Simplify the resulting equation

  4️⃣ Check if the equation holds true
• The solution y = e^(-x)(C1 cos(x) + C2 sin(x)) is valid for the differential equation y'' + 4y' + 3y = 0.


  True
• Match the type of differential equation with the verification strategy:

  First-order DE ↔️ Find the first derivative

  Second-order DE ↔️ Find first and second derivatives
• If the simplified equation is always true, the proposed solution is correct
• Substitution is the key technique for verifying solutions to both first-order and second-order differential equations.

  True
• The key technique used to verify solutions for differential equations is substitution
• What is the first step in verifying a solution for a first-order differential equation?

  Find the derivative