7.2 Verifying Solutions for Differential Equations

• A solution to a differential equation is a function
• A solution to a differential equation must make the equation true when substituted
• The arbitrary constant in the solution y = x^{2} + C</latex> is represented by the letter C
• The derivative of $x^{2} +$$C$ is $2x$
• Substituting a function into a differential equation helps verify if the function is a solution
• One step in substituting a function into a differential equation is to find its derivatives
• Steps to substitute a function into a differential equation:

  1️⃣ Find derivatives

  2️⃣ Substitute into the equation

  3️⃣ Simplify the equation
• Simplifying the equation after substitution helps confirm if the function is a solution
• Combining like terms is a step in simplifying the equation
• What is a solution to a differential equation?

  A function that makes it true
• Substituting a function into a differential equation helps verify if the function is a solution
• Steps to substitute a function into a differential equation

  1️⃣ Find derivatives

  2️⃣ Substitute into the equation

  3️⃣ Simplify
• The function $y =$$e^{ - 2x}$ is a solution to $\frac{dy}{dx} +$$2y =$$0$
• What is the first step in simplifying an equation after substitution?

  Combine like terms
• Combining like terms is a key step in simplifying a differential equation after substitution
• To check if an equation holds true after simplifying, compare both sides
• Steps to check if an equation holds true

  1️⃣ Simplify the equation

  2️⃣ Compare both sides

  3️⃣ Conclude if equation holds true
• What is the primary goal of verifying a solution to a differential equation?

  To check if it holds true
• For exponential differential equations, simplification often involves using exponential identities
• The function $y =$$Ce^{ - 3x}$ is a solution to $\frac{dy}{dx} +$$3y =$$0$
• Match the differential equation type with its verification process

  Linear ↔️ Substitute and simplify

  Exponential ↔️ Use exponential identities